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Chapter V: Mathematical Space and Time

Are points fictions?

Physical Space and Time are thus one with mental space and time, or, more strictly, portions of the one Space and Time may be enjoyed and are identical with parts of the one contemplated Space and Time. Space and Time as we have regarded them are empirical or experienced extension or duration, though as continua of moments or points they have been described by help of conceptual terms. Are Space and Time so regarded the Space and Time of the mathematicians, and if so, what is the difference between the metaphysics and the mathematics of them? Our answer will be, that directly or indirectly mathematics is concerned with empirical Space and Time, and that, however remote from them mathematics may seem and be, they are never in mathematics torn away from their original. But a difficulty meets us at the outset because of the different conceptions of mathematics entertained by mathematicians themselves. According to some, Space and Time are the absolute or total Space and Time consisting of entities called points or instants. According to others, and they are the more philosophical mathematicians, Space and Time are not extension or duration, not in any sense stuff or substance,1 as Descartes, to all intents and purposes, conceived particular spaces to be, but relations between material things which move in them. This is the relational conception; as distinguished from the absolute conception, which is expressed by Newton in the sentence, “For times and spaces are as it were the places as well of themselves as of all other things” (Princ. Book I. Schol. iv.). The contrast of absolute and relational is as we have seen entirely different from that of absolute and relative. But the relational conception of Space and Time carries with it, especially in its more recent forms, consequences which make it seem almost impossible to affirm that there is only one Space and Time, namely, the empirical one, with which mathematics is concerned. Rather it would seem that empirical Space and Time are but particular examples of constructions of a much wider scope.

The Space and Time of the previous chapters are then empirical. In essentials they are absolute Space and Time, though, regarded merely as constituents of the one reality which is Space-Time, they are purged of the errors which attach to them when they are considered independently of one another. All Space in fact is full of Time and there is no such thing as empty Space; all Time occupies Space and there is no such thing as empty Time. I do not think that the ordinary geometry (Euclidean or other) assumes Space to be divorced from Time. It makes no assumption on this point at all. On the contrary, whenever its purpose is suited it conceives of a figure genetically as traced by movement in time. Nor does mathematics in dealing with Time assume it to exist by itself. It treats Time by itself, though for the most part Time enters as being an element of motion, and consequently so far in space. But to treat Space or Time by themselves is not to assume that they can exist apart from one another. There is nothing in the procedure of geometry (I say nothing of the views of geometers themselves in the history of the subject) which implies that Space is a system of resting points, or Time a uniform flow which has no habitat.

The entities called points, of which Space is composed (or the instants of Time), are, it is said, commonly regarded by mathematicians as fictions. Mathematicians are very prone, indeed, to regard the notions and the methods they employ as fictions, as if they were mere constructions of our minds. The famous writer, R. Dedekind, himself one of the authors of that movement which has made geometry by an immense generalisation a department of arithmetic, pronounces numbers to be “free creations of the human mind.”2 Under these circumstances we may ask ourselves the preliminary question, When is a fiction fictitious? I adopt, for instance, a son who becomes my son by a legal fiction, by a generalisation or extension of the notion of son. How much of this fiction is real and how much unreal? So far as I treat him as a son, exchange with him affection and care and perform certain legal and moral obligations to him, the adopted son is really in the place to me of a son. There is no fiction here. There is only fiction if I strain the relation: if I should, for instance, go on to pretend that he owes his height to me and his wit to my wife, that his colourblindness is traceable to her and may be found in her brother, and a little pit in the skin beside his ear to my father. Here the fiction becomes fictitious. The legal and moral relation of sonship must not be interchanged with the natural one of inheritance. But in other respects the fiction is a true description of the new facts initiated by the adoption.

Isolation of points corrected by conception of continuity.

Now the assumption of points as elements of Space in a continuous series is an attempt to describe in ideal and partly conceptual terms the given or empirical fact of the continuity of Space, that any stretch of space however small is divisible, and that there is no smallest part. So far as the point is thought to be a self-subsistent entity by aggregation of which with other points Space is constituted, the point is fictitious. But such an assumption is not in fact, and never need be, made. On the contrary, the idea of the separate point is a first approximation, which is corrected by the notion of its continuity with other points. This happens even in the ordinary elementary geometry. For more thorough-going or philosophical mathematical analysis the concept of continuity taken over from sensible continuity is deepened into the analysis of continuity, which then supersedes that merely sensible continuity. This analysis is a crowning achievement of mathematical speculation. It has been described recently by Mr. Russell.3 Like infinity, which has been touched upon before, it is not a mere negative but a positive conception. It does not rely upon or refer to any mental incapacity in us, that we are unable to reach an end of the division, but upon a real characteristic of the continuous series. We can think of Space extended indefinitely, said Locke, but not of Space infinite, because, in the happy phrase which contains the substance of Kant's later and more famous discussion of the same problem, we cannot “adjust a standing measure to a growing bulk.”4 In reality, infinite Space precedes any finite space. In the same way Space is not merely infinitely divisible in the sense that its division admits no end, but is in itself infinitely divided in the sense that between any two elements there exists another element; so that no two elements—we may call them points—are next or adjacent. Thus, just as before with the infinite the finite meant defect and the infinite self-subsistence, so here the division into a finite number of parts implies selection from the infinite of parts in the real continuum.

In this way the point which is an unextended entity with a fictitious self-subsistence is brought into conformity with facts by the correction of the conception. The definition of continuity starting with separate points screws them, or squeezes them, up into that closeness which is needed to express the nature of Space. Even this degree of closeness is not enough for the perfect definition of the continuum. But the further criterion which ensures that the series of points shall be not merely ‘compact,’ according to the description given that no two points are next points, but ‘dense,’ is more technical than I can take upon me to reproduce here. The effect of it, however, may be illustrated from the old puzzle of Achilles and the tortoise. In that puzzle the steps taken by the two competitors form two series which do not reach in any number of steps the point at which as a matter of fact Achilles overtakes the tortoise. That point is the limit of the two series. But though the limit is not reached, and is not the end or any member of the two series, it is a point on the actual line of the journey which both Achilles and the tortoise make. The fact that Achilles does overtake the tortoise is the very mark that their course, which has been artificially broken up into discrete lengths, is really continuous.

The mathematical notion of continuity contains no dreaded infinite regress; the infinitude is of the essence of the datum and expresses no repetition of steps upon our part. On the other hand, if it be asked what is there in space-points which makes them continuous, we are asking a different question from the question what is the criterion of their continuity. The answer, if I am right, must be that points are continuous because they are not mere points but are instants as well. It is Time which distinguishes one point from another, but it is Time also which connects them. For the point is really never at rest but only a transition in a motion. Now it is this restlessness of the point which is expressed in terms of Space itself by the criteria of continuity which the mathematician adopts in order to free his points from their apparent isolation and self-dependence. We are brought back to the conclusion that the mathematical notion of continuity as applied to Space or Time is an attempt to render in terms of points or instants their crude original continuity, and carries with it the corrective to the apparent isolation of points and instants. At the same time, it must be insisted that the mere concept of continuity of either points or instants is only adequate to the crude continuity of actual Space or Time when the points are recognised as being intrinsically instants and the instants points; just as the concept of dog can only be adequate to a particular dog when it is embodied in individualising circumstance.

The spatialising of Time.

We may pause, before passing on, to complete the remarks which were made in the last chapter on Mr. Bergson's repugnance to the spatialising of Time. Mental time or durée was, we saw, laid out in space, where Space was understood to be the Space common to both the physical world and the mind. Thus the spatialising of mind or Time, which Mr. Bergson regarded as a common and natural vice, is in fact of the essence of Time and mental life. But the Space which Mr. Bergson fears and regards as the bare form of externality, the dead body into which the world resolves when Time is arrested, is what he supposes the Space of the geometer to be. Now if mathematics understood by Space or Time the Space of absolute rest, or its counterpart and mirror in mere undifferentiated, and inert, inefficacious, Time (and this is what Mr. Bergson is contending against), his fears would be justified. Such Space and Time are abstractions and correspond to nothing real. But, as we have seen, the Space and Time of the mathematician are not such, or at least are not necessarily such, and we are assured that they are not in fact so treated. Space is legitimately considered by itself and Time likewise. But they are not considered as made up of separate parts but as continuous, and their continuity is defined. In like manner motion is not for the mathematician made up of separate positions, but of separate positions corrected by continuity. Mr. Bergson's main concern is with motion, and he rightly insists that motion is a whole and continuous, having in his mind the original continuity which is given empirically and is antecedent to the conceptualisation of it in mathematics. This conceptualisation he mistakes apparently for destruction, and supposes that Space has been reduced to a series of separate points, and Time with it to a series of separate moments. It is true, moreover, that since geometry omits Time from Space there is a certain artificiality in the reconstruction of continuity within Space in purely spatial terms, and there is a corresponding artificiality in the continuity of Time without reference to Space. This arises from the nature of the case, and indicates that mathematics is not, like metaphysics, an ultimate treatment of its subject matter—on which topic more anon. What Mr. Bergson appears to forget is that this science works within its limits, but in doing so does do justice to that very continuity and wholeness of Space and Time, and with them motion, for which he himself is pleading.

The consequence of this misapprehension is visible through the twilight which envelops Mr. Bergson's conception of the relation of Time to Space and, with Space, to matter. No one has rendered such service to metaphysics as he has done in maintaining the claims of Time to be considered an ultimate reality. Moreover, Space is for him generated along with Time. The movement of Time, the swing and impulse of the world, the èlan vital, is also a creation of matter. The two mutually involved processes remind us of the roads upward and downward of his prototype Heraclitus. But with his forerunner this relation is conceived quite naïvely: we are told that the unity of opposites means nothing more with Heraclitus than that opposites were two sides of one and the same process, so that day and night were but oscillations of the “measures” of fire and water.5 With Mr. Bergson, on the other hand, Space is a sort of shadow or foil to Time, and not co-equal. It implies degradation and unreality, relatively to Time. Time remains the unique and ultimate reality. We have seen reason to regard them as so implicated in each other that each is vital to the other's existence. But whether this feature of his doctrine, at once the most interesting for the metaphysician and the most obscure and tantalising, is the outcome of his apparent misapprehension of the purpose and legitimacy of geometry, or the latter misapprehension a consequence of his incomplete analysis of the true relation of Space and Time, I leave undetermined, my aim being not to offer criticism of current or past philosophies, but to indicate where the analysis here maintained differs, whether to my misfortune or not, from a deservedly influential system of thought.

Mathematical and empirical Space.

The points of space and instants of time when interpreted aright are no fictions, in the sense of being fictitious, but the elementary constituents of Space and Time, as arrived at by a process which we have described already as being partly imaginative and partly conceptual. That is we suppose the process of division continued without end, and we think of any space as integrated out of points so as to be a continuum, and thus we use the concept of point. But points though in this way ideal are none the less real. They are not made by our thought but discovered by it. To repeat what has been said before, reality is not limited to sensible constituents but contains ideal and conceptual ones. The back of a solid object which we see in front, the taste of an orange which we feel or see are ideal, but they belong none the less to the real solid and the real orange. Likewise the concept or thought of a dog is as real a constituent of the dog as what makes him a singular thing. It is its structural plan. Like all the objects of our experience, any part of Space contains the two aspects of singularity and universality. It is itself and it follows a law of structure. Points are singular, but they have such structure as becomes a point and are so far universal. In like manner, the figures of the mathematician, straight lines, triangles, conic sections, etc., are discovered by a process of idealisation, by an act of selection from the whole of Space. It is easy enough to recognise that this is the case with the geometrical figures taken apart from Time. For from Space we may select, by an ideal act, what Mr. Bradley calls an ideal experiment, the various geometrical figures. We do so whenever we draw them, and disregard the sensuous or sensible irregularities of our draughtsmanship, idealising these irregularities away. The construction of a parabola is an ideal drawing, or rather an ideal selection of points from Space in conformity with a certain law, expressed in the definition of the parabola. Such construction is in no sense a mere result of abstraction from sensible figures, but a discovery by thought that Space contains the geometrical figures which are thus dissected out of Space. Accordingly, in the history of the subject, geometry has proceeded from very simple figures like triangles to the discovery of more and more complex ones.

In maintaining that geometrical figures are ideal selections from Space, but really parts of that Space, I may be thought guilty of inconsistency with my own principles. Experienced reality I have said is not Space, but Space-Time, of which the constituents are not spaces or times but motions. The geometer himself, it will be urged, treats his figures as the locus of motions. Now in physical reality we find no perfectly triangular or parabolic motions; there can be no meaning in the attempt to select such perfect movements from the motions which actually exist. Something might indeed be said in defence of the attempt: that where we have three intersecting directions we have the triangle. In the end, however, it will be seen that the notion of a direction prolonged into a straight line is incapable of defence if it is supposed to exist in the real world of motions. But in fact the objection taken is not relevant. The mathematician's Space is that Space which we have identified as the framework of real motions. It is within this Space, whose reality has been already maintained as essentially involved in Space-Time, that the mathematician draws his lines and circles and parabolas by an ideal selection. In this sense the figures of the geometer are real constructions, and geometry (and the same thing is true of the numbers of arithmetic) is a basal science of reality. When such figures are thought, the geometer can then proceed to treat his figures as the locus of points moving according to a certain law. But such conceptions in no way commit us to the belief that these movements, elaborated as it were by an afterthought, claim to be selected from the real world of motions.

Not different as conceptual and perceptual;

We can now ask ourselves the question, what is the relation of empirical Space to geometrical Space, and answer it by saying that they are the same, but that geometry treats Space differently from ordinary experience. Mathematical Space and Time are sometimes contrasted as conceptual or intelligible with empirical Space and Time as perceptual. The contrast, in the first place, is not strictly correct.6 For Spaces and Times are not objects of perception as trees or houses are. We have no sense for Space or Time, nor even in the proper meaning of sensation, for movement; they are apprehended through the objects of perception, the things which fill spaces and times, but not by sense. They are more elementary than percepts. Half our difficulties have arisen from attempting to regard Space as given to us by touch or sight instead of only through touch and sight. Hence it is that some have entertained the naïve and impossible notion that geometrical figures are got from material objects by a process of abstraction. They are got, as we have seen, by a selection from Space, which is always an ideal discovery. The only resemblance between figures of empirical Space and percepts is that they are individual or singular. Let us, however, overlook the inaccuracy of holding Space to be sensible at all, because the question is not ripe for our discussion in spite of the danger which such a notion involves that Space may be thought dependent on us in much the same way as colours and touches are supposed to be. It still remains true that the distinction of perceptual and conceptual is not sufficient to distinguish the Space of things from that of geometry. For empirical spaces besides being singular (and perceptual) are also conceptual. Each point is distinct from the other, because it is a point-instant and its time discriminates it; but empirical Space involves also the concept point or point-instant. Its point-instants have a universal character or structure. Like material or sensible empirical things, spaces (and times) are saturated with concepts. On the other hand, the Space of geometry also consists of points, and the figures which it deals with are different instances of figures of one and the same kind. There are various triangles and parabolas. These are the so-called ‘mathematicals’ of Plato, which he regarded (mistakenly as we shall see later) as intermediate between sensible things and universals. There are individual parabolas as well as the universal parabola. Thus empirical Space contains concepts and mathematical Space contains percepts. I am speaking here of the Space of elementary geometry, and am not considering as yet the speculative or arithmetised form of that science.

but differently treated.

The difference lies not in there being a difference of empirical and geometrical Space but in the treatment of it. Geometry treats it wholly conceptually. Though there are many triangles and parabolas and points it considers the universal parabola or circle or point. It deals not with circles but with the circle or any circle. And it is able to do so, and is justified in doing so, because it abstracts from the Time of Space, though it does not as we have so often said exclude it. Conceiving the point without its time, it regards one point as the same as another. But that, even so, it does not exclude the real individuality of the point is evident from the fact that though its parabola has no definite place in space but may be anywhere, yet there is a relative individuality. For supposing the axis and the focus of the parabola fixed, all the other points of the curve are fixed in relation to it.

Geometry deals with figures, not Space.

This leads us on to a more significant point of difference which in the end is identical with the one we have mentioned. Geometry omits Time from its Space, or introduces it again by a quasi-spatial artifice in the use of a fourth co-ordinate, the time co-ordinate, and consequently it treats its Space conceptually. But geometry is in strictness not concerned with Space as such at all; that is the office of metaphysics. Geometry is concerned with figures in Space; its subject matter is the various empirical or varying determinations of that a priori material, Space. It is the empirical science of such figures which are its data, which accordingly, like any other empirical science, it attempts to weave into a consistent system. Metaphysics, on the other hand, is not a science of empirical figures in Space. But one of its problems is what is the nature of Space and how there are figures within it. In like manner, arithmetic is not concerned with number as such, but with the empirical numbers (of all kinds) which are discoverable within the region of number, as empirical or varying determinations of that a priori material. The mathematician is not as a mathematician concerned with these ultimate questions; he is only concerned with them, by the interchange of friendly offices between metaphysics and the special sciences, by which the special sciences have been enabled to contribute so helpfully to metaphysics; because the student of a special department may also, if he has the eye for its ultimate questions, approach them with a fulness of knowledge. He may at the same time view these ultimate problems for his own purposes in a different light from the metaphysician, and this we shall see to be the case with the mathematician.7 Now, just because the metaphysician deals with Space and number as such, it is of prime importance for him that individual points and circles are different from each other. But geometry not dealing with the problem of the individuality of its points and circles concerns itself with points and circles as such, and thus becomes wholly conceptual.

There are thus not two Spaces, the Space of elementary geometry and empirical Space, but one Space considered in metaphysics and mathematics with a different interest. The interest of mathematics is in the figures which are the empirical variations of the a priori Space; the interest of metaphysics is in the nature of Space itself. The question may be asked, How can a point or rather a point-instant be individual, each one different from all others, as metaphysics insists, and yet a point-instant be a universal? What makes the difference between the universal and its particulars? We have not yet reached the stage at which this inquiry can be answered. We shall see that the very difference of universal and particular depends on the fact that each point-instant is itself, and yet of the same character as others. At present it is enough to observe that the elementary universal, point-instant, is comparable to a proper name like John Smith, the whole meaning of which, as Mr. Bosanquet has said, is to indicate any particular individual; so that while any number of persons have that name, the name does not so much imply properties which are common to them all, but merely designates in each instance of its use a single individual, and is thus used in a different sense with each. I need hardly stop to reject the supposition that a point-instant is as it were the meeting-place of two concepts, point and instant, as if a combination of two concepts could confer individuality. For, firstly, no combination of concepts makes an individual. Secondly, point and instant are not concepts combined to make that of point-instant, as hard and yellow are combined in gold. For point and instant are not separable from one another, but each implies the other, and the concepts point and instant are merely elements distinguished in the concept point-instant. But all these matters belong strictly to a later stage of our inquiry. They are mentioned here only to anticipate difficulty.

Space and geometries of it.

The starting-point of geometry is then empirical Space presented in experience as what can only be described in conceptual terms as a continuum of points. The elaborate analysis of continuity by the speculative mathematicians does but explain what is given in this empirical form. But when I go on to ask what more precisely geometry does, and have regard to the history of the various geometries and to the most recent reduction of geometry to the status of an illustration of algebra, I find myself in danger of the fate which is said to overtake those who speak of mathematics without being mathematicians.8 I have to do what I can, and I hope without presumption, with such information as is open to me. My object is the modest one of setting the empirical method of metaphysics as occupied with spaces and numbers in its relation first to elementary geometry of three dimensions, and next to the more generalised conceptions of mathematical procedure for which geometry is but a special application of arithmetic, or rather both geometry and arithmetic fall under one science of order.

Starting then with empirical Space, geometry, like any other science, proceeds by means of axioms, definitions, and postulates, to discover what may be learnt about figures in space and, in general, about spatial relations. Thus Euclid from his premisses arrives at properties of triangles. The axioms and postulates of geometry are its hypotheses. Even the assumption of points when they are given a semi-independent existence in order to give support to the imagination is hypothetical. But Euclid's axioms are not the only ones out of which a body of geometrical truths can be constructed which still are applicable to empirical Space. There are many geometries though there is but one Space. Strictly speaking, it is only by a mistake of language that we speak of non-Euclidean Space or even of Euclidean Space; we have only Euclidean or non-Euclidean geometry. In the first place, while Euclidean geometry is metrical and involves magnitude and measurement, there is the more abstract geometry of position, or projective geometry, “which involves only the intersectional properties of points, lines, planes, etc.,”9 and in metric geometry there are the modern systems which introduce notions of order or motion. But besides these there are the geometries, still three-dimensional, which are not Euclidean at all. The late H. Poincaré, as is well known, thought that it was impossible and indeed meaningless to ask whether Euclid or these other geometries were true. They differ not in respect of truth but of practical convenience. Euclid's is the most convenient. It is by no means involved in empirical Space that a straight line should be the shortest between two points. Poincaré imagines a spherical world where the temperature changes from centre to circumference, and bodies shrink or grow with the fall or rise of the temperature. Apparently such a geometry (in which the shortest lines are circles) would apply to empirical Space “within the possible error of observation.”10 In other words, the difference of its conclusions from those of Euclidean geometry would not be capable of detection by our instruments.

This also, I understand, applies to the non-Euclidean geometries of Lobatchewsky and others, the so-called hyperbolic and elliptic geometries. Now, in the case of these geometries, the question does not arise whether they take us into a world different from our experienced Space. They are merely different systems of explaining, not the ultimate nature of Space, but its behaviour in detail. They employ different postulates. At the same time they introduce us to another feature of geometry and of mathematics generally, its method of generalisation. Euclidean geometry is only one instance of geometry of empirical Space. In it, a parallel may be drawn through any point outside a straight line to that line, and only one. In hyperbolic geometry there are two parallels; in elliptic geometry none. Or we may put the matter differently by reference to what is called the space-constant, or to what is less accurately spoken of as the ‘curvature’ of the Space. This constant has a finite value, positive or negative, in the other two geometries; in Euclidean geometry it tends to infinity. In the less accurate language the curvature of Euclidean space is zero, in the other two cases it is positive or negative. Now it. is the generalising tendency of mathematics which has led ultimately to the reduction of geometry to arithmetic, and it raises the question in what sense the world of mathematical entities so conceived is real, whether it is not a neutral world, and empirical geometry only an application of its laws to sensible material.11

Generalised ‘Space’.

The simplest though not the most important example of such generalisation is found in geometries of more than three dimensions. Dimensionality, as Mr. Young points out,12 is an idea of order. A point by its motion generates a straight line, a line a plane, a plane solid Space, and this exhausts all the points of Space. We may think then of Space as a class of points arranged in three orders or dimensions. But this notion, once drawn from empirical Space, may be extended or generalised; and we may think of a class of any number of dimensions which will have its geometry or be a new so-called ‘Space.’ We have taken a notion and generalised it, cutting it loose as it were from its attachments to the one empirical Space. Such generalisation is of the very life of mathematics, and its most important example is the process by which the notion of number has been extended. The study of numbers begins with the integral numbers, however they are conceived; but the notion of number has been extended by successive steps, so that there have been included in the number-system fractions as well as integral rational numbers, negative as well as positive numbers, irrationals, imaginary numbers, complex numbers consisting of a rational combined with an irrational number, and now the class of infinite or transfinite numbers. All these symbols have been so defined as to preserve the general laws of ordinary numbers, and great advances in the understanding of numbers have been marked by successful definitions, like the famous definition of an irrational number by R. Dedekind. To a certain extent it may be sufficient to describe these numbers as conventional, but that they are not mere conventions is shown partly by their having been suggested in some cases by geometry (as incommensurable numbers, for example, by the relation of magnitude between the side and the diagonal of a square); partly by the possibility of interpreting them geometrically. Mr. Young quotes a saying of Prof. Klein, that it looks as though the algebraical symbols were more reasonable than the men who employed them.13

Now I assume that the notion of an n-dimensional geometry is fruitful and profitable as a topic of inquiry. And if so, it seems to me to be as unreasonable to deny the value of it, which some philosophers are inclined to do, as it would be to reject imaginaries because there are no imaginary points in real Space. What we have in both cases alike is the investigation of certain notions for their own sakes when taken apart from their attachments; and the question rather is not whether they are legitimate, for I do not see how their legitimacy can be questioned, but the much more interesting question of whether they ever lose their original connection with the empirical so as to constitute a ‘neutral’ world of thought which is neither physical nor mental.

A product of art.

This question, so far as it is raised, even at this stage appears to admit of an answer. The idea of dimensionality taken by itself is combined with that of number, and a system is constructed by thought of elements in an n-dimensional total, and the consequences are worked out of this assumption. The systems are not, properly speaking, Spaces at all, nor their elements points in the empirical sense, but three-dimensional Space may be treated as derived from such a ‘Space,’ e.g. from four-dimensional ‘Space,’ on the analogy of the derivation of two-dimensional Space (which after all is an abstraction) from three-dimensional Space.. Now if we may assume for the moment what will appear later,14 that integral number itself is but a conception founded in empirical Space-Time, what we have here is nothing more in kind than the imagination of a gold mountain or any other work of imagination, only that in imagination the elements are sensory and found in the sensory world, whereas thought liberates itself from this condition. If the notions of dimensionality and number are rooted in Space-Time, the construction of a more than three-dimensional ‘Space’ does not lead us into a neutral world but takes notions which are empirical at bottom and combines them by an act of our minds. But just as the arbitrary act of imagination by which we construct a chimaera leaves us still dealing with physical features, though combined in a way which is not verified in physical fact, so in these thought constructions we are dealing all the time with ideas belonging to the empirical world. No new or neutral world is established, but the freedom of thought gives rise to fresh combinations.

No one would admit that a chimaera belongs to a neutral world; but rather so far as it claims to be real its claims are a pretence. The question then we have to ask is, are such intellectual constructions as many-dimensional ‘Spaces,’ or imaginary numbers, merely imaginary or are they true. A chimaera is not true, though it may have its place in a world of art as a work of pure imagination; and it is not true because it does not follow the lines of nature in the organic world. But a concept which is founded in the nature of Space-Time may admit of extensions or generalisations which are the work of pure thought, and discovered by it, and yet being on the lines of nature in the empirical world of Space-Time may be coherent with the spatial or numerical system and, at whatever degree of remoteness, be applicable again to the nature from which it sprang. Thus, to take an instance which supposes very little acquaintance with geometry, the idea that all circles pass through the same two imaginary points at infinity is a pure construction of thought. It is founded on the general proposition that two curves of the second degree intersect each other in four points. Two intersecting circles also intersect at these circular points at infinity. But by the use of this intellectual construction we can pass by projection from properties of the circle to properties of the ellipse. Such intellectual constructs are thus not mere exercises of thought, like a chimaera, but are coherent with the system of thoughts which have correspondents in real Space. They have therefore a double value, first in themselves, and secondly in the application of them.

Now as to the various kinds of numbers which have been discovered and introduced into arithmetic in virtue of the tendency towards that generality which, Mr. Whitehead says, the mathematician is always seeking,15 their connection with the integral numbers has been observed above, where I have mentioned the fact that they admit of spatial interpretation. As to the usefulness of the extensions of the notion of dimensionality, I can but quote the words of Mr. Young (p. 174): “It may be stated without fear of contradiction that the study of such spaces has been of the greatest practical value, both in pure mathematics and in the applications of mathematics to the physical sciences.”

Thus in one respect the extensions in which geometry deserts empirical Space and creates new ‘Spaces,’ or the constructions within the system of number, are, it would seem, comparable to the scaffoldings by the help of which we build great buildings or ships. They allow us to raise the structure with which we are concerned, and to come indirectly into contact with it. Sometimes, as in the scaffolding of a building, parts of the scaffolding may be inserted into the building itself which is being raised. Sometimes they may be detached like the great framework of steel within which a ship is built, such as one sees as one steams down the river at Belfast or other great dockyard. These are still material structures, and belong to the same world as the ships or buildings. In higher geometry or arithmetic we have in like manner works of art whose materials are derived from experienced Space-Time, however intricately combined by thought, and they also have their utility in their application. But in another respect the comparison is faulty. For the scaffoldings of houses and ships exist only in order to build houses or ships. But the mathematician's constructions are made for their own sakes and are discoveries within geometry and arithmetic itself; like all scientific constructions they have a value irrespective of utility. Still, also like them, they are based upon and draw their life from the empirical material with which they are in organic connection.

Relation of the generalisations to empirical Space and Time.

I am far from supposing that the notion of many-dimensional ‘Spaces’ is comparable in importance philosophically with the numerical constructions which have given us in arithmetic the irrational, imaginary, or transfinite, numbers. I am not able to judge. But it seems fairly clear that the intimacy of connection between the first set of constructions and the empirical world is much less than in the case of the second set. A four-dimensional ‘Space’ is not a Space at all, and it appears to be rather a means for discovery in three-dimensional Space than itself the discovery of something in the world of Space; rather a work of art than a discovery. But the numbers are discoveries within the system of numbers. My object, however, has not been to assign to these different constructions their grades of value; it has been no more than to indicate that they do not take us into a neutral world of thought but keep us still in contact with the one Space and Time which we apprehend in experience, and seek to understand in mathematics in their empirical determinations by the selective analysis and intellectual construction employed in mathematics. In other words, we are not entitled to say, because by generalisation we arrive at a world of thoughts, that that world of thought is for metaphysics a neutral world of which our empirical world is the manifestation under certain conditions of sensible experience; that empirical Space, for instance, is a particular example of a system of complex numbers. We have in fact started from the empirical world itself, in particular from empirical Space and Time, extended by thought the conceptions derived from it, and descended again to empirical Space. The procedure is legitimate, but it does not establish the primacy of a neutral world. Metaphysically, empirical Space and Time are themselves the foundation of this neutral world. There is however another problem set to us, which belongs to the theory of knowledge, that is to that chapter of metaphysics. We have to ask what kind of reality belongs to such thought-constructions, and this runs into the general question, what reality belongs to ideas and to hypotheses and all assumptions and to mere imaginations and illusions, and ideas which are commonly called unreal, like a round square, which still remain objects of thought or we could not speak about them. Now it is only one solution that there is a world of neutral being simpler than the world of physical or mental things which exist. There may be the world of truth and error or art (suggested above) which is not to be characterised as neither physical nor mental but as both physical and mental.16 For our present purpose it is enough to insist that metaphysically all these constructions are rooted in the empirical world of existence, and ultimately in empirical Space and Time.

But in order properly to understand what is implied in the generalisation by which geometry and arithmetic become one science, we must go further and discuss a fundamental question which has been reserved. Dimensionality is an idea of order and order is connected with relation. I have assumed provisionally that for empirical metaphysics order and number can be exhibited as derived from Space-Time and dependent on it. But we can only satisfy ourselves that the concepts of mathematics are still attached to empirical Space-Time by examining the view that Space and Time are relations and not as we have supposed a stuff. We shall then see that the concepts in which mathematics appears to move away from Space-Time are in the end saturated with the notion of Space-Time. We have thus to ask ourselves, what are relations in Space and Time, and under what conditions Space and Time can be treated as systems of relations.

  • 1.

    See, for the qualification of the use of the word substance, Bk. II. ch. x. p. 341: Space-Time is not substance at all, but stuff.

  • 2.

    Was sind und was sollen die Zahlen. Preface, p. vii. (Brunswick, 1911, ed. 3).

  • 3.

    B. Russell, Our Knowledge of the External World (Chicago and London, 1914), ch. v.

  • 4.

    Essay, Bk. II. ch. xvii. § 7.

  • 5.

    J. Burnet, Early Greek Philosophy (London, 1908, ed. 2), p. 186.

  • 6.

    In an article, ‘What do we mean by the question: Is our Space Euclidean?’ in Mind, N. S. vol. xxiv. p. 472, Mr. C. D. Broad remarks similarly upon this distinction; though not to the same purpose.

  • 7.

    The possible helpfulness of metaphysics, within its limitations, to the special sciences has not so generally been recognised by them.

  • 8.

    “On the other hand,” says Mr. A. N. Whitehead (Introduction to Mathematics, p. 113), “it must be said that, with hardly any exception, all the remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it are entirely worthless, being either trivial or wrong.” He is pointing the contrast with Descartes.

  • 9.

    Fundamental Concept of Algebra and Geometry, by J. Wesley Young, p. 135 (New York, 1911), to which I am deeply indebted in what follows for information.

  • 10.

    J. W. Young, loc. cit. p. 23.

  • 11.

    “The geometrical system constructed upon these foundations (i.e. those of Lobatchewsky and Bolyai) is as consistent as that of Euclid. Not only so, but by a proper choice of a parameter entering into it, this system can be made to describe and agree with the external relations of things” (H. S. Carslaw, Elements of Non-Euclidean Plane Geometry and Trigonometry, London, 1916).

  • 12.

    Loc. cit. p. 170.

  • 13.

    P. 112.

  • 14.

    It is a great disadvantage for me that I cannot anticipate the discussion of this point. Without it my assertion may appear to be dogma. See Book II. where number is described (as well as order) along with the other categories in its place.

  • 15.

    Introduction to Mathematics, p. 82.

  • 16.

    See later, Bk. III. ch. ix. A.