CHAPTER L¶
Part VI — Space — The Continuity of Space (§§ 416–422, pp. 437–444)
§ 416. The continuity of a projective space¶
416. It has been commonly supposed by philosophers that the continuity of space was something incapable of further analysis, to be regarded as a mystery, not critically inspected by the profane intellect. In Part V, I asserted that Cantor’s continuity is all that we require in dealing with space. In the present chapter, I wish to make good this assertion, in so far as is possible without raising the question of absolute and relative position, which I reserve for the next chapter.
Let us begin with the continuity of projective space. We have seen that the points of descriptive space are ordinally similar to those of a semi-continuous portion of a projective space, namely to the ideal points which have real correlatives. Hence the continuity of descriptive space is of the same kind as that of projective space, and need not, therefore, be separately considered. But metrical space will require a new discussion.
It is to be observed that Geometries, as they are treated now-a-days, do not begin by assuming spaces with an infinite number of points; in fact, space is, as Peano remarks [1], a word with which Geometry can very easily dispense. Geometries begin by assuming a class-concept point, together with certain axioms from which conclusions can be drawn as to the number of points. So, in projective Geometry, we begin with the assumption that there are at least two points, and that any two points determine a class of points, the straight line, to which they and at least one other point belong. Hence we have three points. We now introduce the new assumption that there is at least one point not on any given straight line. This gives us a fourth point, and since there must be points on the lines joining it to our previous points, we obtain three more points—seven in all. Hence we can obtain an infinite denumerable series of points and lines, but we cannot, without a further assumption, prove that there are more than three points on any one line. Four points on a line result from the assumption that, if b and d be harmonic with respect to a and c, then b and d are distinct. But in order to obtain an infinite number of points on a line, we need the further assumptions from which the projective order results [2]. These assumptions necessitate a denumerable series of points on our line. With these, if we chose, we might be content. Such a series of points is obtained by successive quadrilateral constructions; and if we chose to define a space in which all points on a line could be obtained by successive quadrilateral constructions starting with any three points of the line, no contradiction would emerge. Such a space would have the ordinal type of the positive rationals and zero: the points on a line would form a compact denumerable series with one end. The extension, introduced by assuming that the series of points is continuous, is only necessary if our projective space is to possess the usual metrical properties—if, that is to say, there is to be a stretch, with one end and its straight line given, which is to be equal to any given stretch. With only rational points, this property (which is Euclid’s postulate of the existence of the circle) cannot hold universally. But for pure projective purposes, it is irrelevant whether our space possesses or does not possess this property. The axiom of continuity itself may be stated in either of the two following forms. (1) All points on a line are limits of series of rational points, and all infinite series of rational points have limits; (2) if all points of a line be divided into two classes, of which one wholly precedes the other, then either the first class has a last term, or the last has a first term, but both do not happen. In the first of these ways, the continuity which results is exactly Cantor’s, but the second, which is Dedekind’s definition, is a necessary, not a sufficient, condition for Cantor’s continuity. Adopting this first definition, the rational points, omitting their first term, form an endless compact denumerable series; all points form a perfect series; and between any two points there is a rational point, which is precisely the ordinal definition of continuity [3]. Thus if a projective space is to have continuity at all, it must have the kind of continuity which belongs to the real numbers.
§ 417. The continuity of a metrical space¶
417. Let us consider next the continuity of a metrical space; and, for the sake of definiteness, let us take Euclidean space. The question is here more difficult, for continuity is not usually introduced by an axiom ad hoc, but appears to result, in some sense, from the axioms of distance. It was already known to Plato that not all lengths are commensurable, and a strict proof of this fact is contained in the tenth book of Euclid. But this does not take us very far in the direction of Cantor’s continuity. The gist of the assertion that not all lengths are commensurable, together with the postulate of the circle, may be expressed as follows. If AB, AC he two lengths along the same straight line, it may happen that, if AB be divided into m equal parts, and AC into n equal parts, then, however m and n may be chosen, one of the parts of AB will not be equal to one of the parts of AC, but will be greater for some values of m and n, and less for others; also lengths equal to either may be taken along any given line and with any given end-points [4]. But this fact by no means proves that the points on a line are not denumerable, since all algebraic numbers are denumerable. Let us see, then, what our axioms allow us to infer.
In Greek Geometry there were two great sources of irrationals, namely, the diagonal of a square and the circumference of a circle. But there could be no knowledge that these are irrationals of different kinds, the one being measured by an algebraic number, the other by a transcendent number. No general method was known for constructing any assigned algebraic number [5], still less for constructing an assigned transcendent number. And so far as I know, such methods, except by means of limits, are still wanting. Some algebraic and some transcendent numbers can be constructed geometrically without the use of limits, but the constructions are isolated, and do not follow any general plan. Hence, for the present, it cannot be inferred from Euclid’s axioms that space has continuity in Cantor’s sense, or that the points of space are not denumerable. Since the introduction of analytic Geometry, some equivalent assumption has been always tacitly made. For example, it has been assumed that any equation which is satisfied by real values of the variables will represent a figure in space; and it seems even to be universally supposed that to every set of real Cartesian coordinates a point must correspond. These assumptions were made, until quite recent times, without any discussion at all, and apparently without any consciousness that they were assumptions.
When once these assumptions are recognized as such, it becomes apparent that, here as in projective space, continuity must be introduced by an axiom ad hoc. But as against the philosophers, we may make the following remark. Cantor’s continuity is indubitably sufficient to satisfy all metrical axioms, and the only question is, whether existent space need have continuity of so high an order. In any case, if measurement is to be theoretically possible, space must not have a greater continuity than that of the real numbers.
The axiom that the points on a line form a continuous series may be put in the form which results from amending Dedekind, or in the form that a line is a perfect series. In the first form, every section of the line is definable by a single point, which is at one end of one of the parts produced by the section, while the other part has no end. In the second form, which is preferable because, unlike the first, it completely defines the ordinal type, every infinite series of points has a limit, and every point is a limiting point. It is not necessary to add that the line has cohesion [6], for this results from the axioms of Archimedes and of linearity, which are in any case essential to measurement. Whether the axiom of continuity be true as regards our actual space, is a question which I see no means of deciding. For any such question must be empirical, and it would be quite impossible to distinguish empirically what may be called a rational space from a continuous space. But in any case there is no reason to think that space has a higher power than that of the continuum.
§ 418. An axiom of continuity enables us to dispense with the postulate of the circle¶
418. The axiom of continuity enables us to dispense with the postulate of the circle, and to substitute for it the following pair. (1) On any straight line there is a point whose distance from a given point on the line is less than a given distance. (2) On any straight line there is a point whose distance from a given point on or off the line is greater than a given distance. From these two assumptions, together with continuity, the existence of the circle can be proved. Since it is not possible, conversely, to deduce continuity from the circle, and since much of analytic Geometry might be false in a discontinuous space, it seems a distinct advance to banish the circle from our initial assumptions, and substitute continuity with the above pair of axioms.
§ 419. Is space prior to points?¶
419. There is thus no mystery in the continuity of space, and no need of any notions not definable in Arithmetic. There is, however, among most philosophers, a notion that, in space, the whole is prior to the parts [7]; that although every length, area, or volume can be divided into lengths, areas, or volumes, yet there are no indivisibles of which such entities are composed. According to this view, points are mere fictions, and only volumes are genuine entities. Volumes are not to be regarded as classes of points, but as wholes containing parts which are never simple. Some such view as this is, indeed, often put forward as giving the very essence of what should be called continuity. This question is distinct from the question of absolute and relative position, which I shall discuss in the following chapter; For, if we regard position as relative, our present question will arise again concerning continuous portions of matter. This present question is, in fact, essentially concerned with continuity, and may therefore be appropriately discussed here.
The series which arise in Arithmetic, whether continuous or not, are essentially composed of terms—integers, rationals, real numbers, etc. And where we come near to the continuity of space, as in the case of the real numbers, each real number is a segment or infinite class of rationals, and no denial that a segment is composed of elements is possible. In this case, we start from the elements and gradually construct various infinite wholes. But in the case of space, we are told, it is infinite wholes that are given to begin with; the elements are only inferred, and the inference, we are assured, is very rash. This question is in the main one of Logic. Let us see how the above view is supported.
Those who deny indivisible points as constituents of space have had, in the past, two lines of argument by which to maintain their denial. They had the difficulties of continuity and infinity, and they had the way in which space is presented in what, according to their school, they called intuition or sensation or perception. The difficulties of continuity and infinity, as we saw in Part V, are a thing of the past; hence this line of argument is no longer open to those who deny points. As regards the other argument, it is extremely difficult to give it a precise form—indeed I suspect that it is impossible. We may take it as agreed that everything spatial, of whose existence we become immediately aware in sensation or intuition, is complex and divisible. Thus the empirical premiss, in the investigation of space, is the existence of divisible entities with certain properties. But here it may be well to make a little digression into the meaning of an empirical premiss.
§ 420. Empirical premisses and induction¶
420. An empirical premiss is a proposition which, for some reason or for no reason, I believe, and which, we may add, is existential. Having agreed to accept this proposition, we shall usually find, on examination, that it is complex, and that there are one or more sets of simpler propositions from which it may be deduced. If P be the empirical premiss, let A be the class of sets of propositions (in their simplest form) from which P may be deduced; and let two members of the class A be considered equivalent when they imply one another. From the truth of P we infer the truth of one set of the class A. If A has only one member, that member must be true. But if there are many members of the class A, not all equivalent, we endeavour to find some other empirical premiss $P'$, implied by all sets of simple propositions of the class $A'$. If now it should happen that the classes A and $A'$ have only one common member, and the other members of A are inconsistent with the other members of $A'$, the common member must be true. If not, we seek a new empirical premiss $P''$, and so on. This is the essence of induction [8]. The empirical premiss is not in any essential sense a premiss, but is a proposition which we wish our deduction to arrive at. In choosing the premisses of our deduction, we are only guided by logical simplicity and the deducibility of our empirical premiss.
§ 421. There is no reason to desire our premisses to be self-evident¶
421. Applying these remarks to Geometry, we see that the common desire for self-evident axioms is entirely mistaken. This desire is due to the belief that the Geometry of our actual space is an à priori science, based on intuition. If this were the case, it would be properly deducible from self-evident axioms, as Kant believed. But if we place it along with other sciences concerning what exists, as an empirical study based upon observation, we see that all that can be legitimately demanded is that observed facts should follow from our premisses, and, if possible, from no set of premisses not equivalent to those which we assume. No one objects to the law of gravitation as being not self-evident, and similarly, when Geometry is taken as empirical, no one can legitimately object to the axiom of parallels—except, of course, on the ground that, like the law of gravitation, it need only be approximately true in order to yield observed facts. It cannot be maintained that no premisses except those of Euclidean Geometry will yield observed results; but others which are permissible must closely approximate to the Euclidean premisses. And so it is with continuity: we cannot prove that our actual space must be continuous, but we cannot prove that it is not so, and we can prove that a continuous space would not differ in any discoverable manner from that in which we live.
§ 422. Space is an aggregate of points, not a unity¶
422. To return from this digression: we agreed that the empirical premisses, as regards the continuity of space, are concerned always with divisible entities which have divisible parts. The question before us is whether we are to infer from this that the logical premisses for the science of existing space (i.e. the definition of existing space) may or must be concerned with divisible entities. The question whether our premisses must be concerned with divisible entities is fully answered, in the negative, by actual Geometry, where, by means of indivisible points, a space empirically indistinguishable from that in which we live is constructed. The only reasons hitherto alleged by philosophers against regarding this answer as satisfactory, are either such as were derived from the difficulties of infinity and continuity, or such as were based upon a certain logical theory of relations. The former have been already disproved; the latter will be discussed in the next chapter. The question whether our premisses may be concerned with divisible entities is far more difficult, and can be answered only by means of the logical discussions of Part II. Whatever is complex, we then decided (§143), must be composed of simple elements; and this conclusion carries us a long way towards the decision of our present question. But it does not quite end our doubts. We distinguished, in Part II, two kinds of wholes, namely aggregates and unities. The former may be identified, at any rate for present purposes, with classes, while the latter seem to be indistinguishable from propositions. Aggregates consist of units from whose addition (in the sense presupposed in Arithmetic) they result; unities, on the contrary, are not reconstituted by the addition of their constituents. In all unities, one term at least is either a predicated predicate or a relating relation; in aggregates, there is no such term. Now what is really maintained by those who deny that space is composed of points is, I imagine, the view that space is a unity, whose constituents do not reconstitute it. I do not mean to say that this view is consciously held by all who make the denial in question, but that it seems the only view which renders the said denial reasonable.
Before discussing this opinion, it is necessary to make a distinction. An aggregate may be an aggregate of unities, and need by no means be an aggregate of simple terms. The question whether a space is an aggregate of unities or of simple terms is mathematically, though not philosophically, irrelevant; the difference of the two cases is illustrated by the difference between an independent projective space and the projective space defined in terms of the elements of a descriptive space. For the present, I do not wish to discuss whether points are unities or simple terms, but whether space is or is not an aggregate of points.
This question is one in which confusions are very liable to occur, and have, I think, actually occurred among those who have denied that a space is an aggregate. Relations are, of course, quite essential to a space, and this has led to the belief that a space is, not only its terms, but also the relations relating them. Here, however, it is easy to see that, if a space be the field of a certain class of relations, then a space is an aggregate; and if relations are essential to the definition of a space, there must be some class of relations having a field which is the space. The relations essential to Geometry will not hold between two spatially divisible terms: there is no straight line joining two volumes, and no distance between two surfaces. Thus, if a space is to be defined by means of a class of relations, it does not follow, as is suggested, that a space is a unity, but rather, on the contrary, that it is an aggregate, namely the field of the said class of relations. And against any view which starts from volumes or surfaces, or indeed anything except points and straight lines, we may urge, with Peano [9], that the distinction between curves, surfaces, and volumes, is only to be effected by means of the straight line, and requires, even then, the most elaborate developments [10]. There is, therefore, no possibility of any definite Geometry without points, no logical reason against points, and strong logical reasons in their favour. We may therefore take it as proved that, if we are to construct any self-consistent theory of space, we must hold space to be an aggregate of points, and not a unity which is indefinable as a class. Space is, in fact, essentially a class, since it cannot be defined by enumeration of its terms, but only by means of its relation to the class-concept point. Space is nothing but the extension of the concept point, as the British army is the extension of the concept British soldier; only, since the number of points is infinite, Geometry is unable to imitate the Army-List by the issue of a Space-List.
Space, then, is composed of points; and if analytical Geometry is to be possible, the number of points must be either equal to, or less than, the number of the continuum. If the number be less, some propositions of the accepted Geometry will be false; but a space in which the number of points is equal to the number of finite numbers, and in which the points of a line form a series ordinally similar to the rationals, will, with suitable axioms, be empirically indistinguishable from a continuous space, and may be actual. Thus Arithmetic, as enlarged by Cantor, is undoubtedly adequate to deal with Geometry; the only question is, whether the more elaborate parts of its machinery are required. It is in number that we become certain of the continuum; among actual existents, so far as present evidence shows, continuity is possible, but cannot be rendered certain and indubitable.
Author’s Footnotes (1903 Edition)