The Analysis of Matter

in (i.e. the class of members of members of ), and their product, i.e. the events which belong to every member of . Then is not null, because is contained in it, and , are connected (in virtue of the definition of collinearity). Let consist of all the 's except , of all the except , etc. Let be the events belonging to all members of and generally let be the events belonging to all members of ; and let be the sum of all the 's. Then consists of all those events which belong to all sufficiently late 's; i.e. to say that an event is a member of is to say that there is an such that the event is a member of for all values of . It will be observed that is contained in , therefore is contained in . It follows that, if , are two members of , there is an such that , are both members of . Hence they are both members of . Hence any five members of are co-punctual, and therefore there is at least one point which contains the whole of , since is contained in . If there is a limit, say , to the series of 's, we require: (1) That should be beyond all the 's, i.e. that for every and we should have contained in i.e. that we should have contained in ; (2) That there should be no point beyond all the 's but between them and , i.e. that, if is any point such that is contained in , then is contained in . [Pg 310] A sufficient condition is, therefore, . If there is a point fulfilling this condition, it is the required limit. If there is an event such that every quartet of is co-punctual with and every quartet of which is co-punctual with is a part of , then there is a point which contains and has for a member, and this point will be such that , so that it will be the required limit. But if there is no such event as , we must proceed differently. In this case we need a new axiom, namely: If is between and , and is a member of but not of , then there is a quartet which is contained in and but is not co-punctual with . In the figure, represents a member of such a quartet. Given this axiom, we proceed as follows. Since is between and , if is a member of but not of , there is a quartet which is contained in and , but is not co-punctual with . Now is contained in ; therefore there is a quartet which is a part of but is not co-punctual with . It follows by transposition that if is a member of and every quartet of is co-punctual with , then is a member of . It follows that is a member of , , ... so that is a member of . Hence, since may be any member of , it follows that any member of which is co-punctual with the whole of is a member of . Now the terms co-punctual with the whole of constitute the class . Hence the common part of and is contained in , and is therefore equal to , since is contained in and in . [Pg 311] Now if is a point which contains , it follows that is contained in ; hence is contained in , and is therefore equal to , since is contained in and in . Hence is the required limit. It follows from this that a compact series of points contained within a stretch of collinear points is continuous. It does not follow that there are compact series of points; this would require existence-axioms which there is no object in introducing, since we do not know whether space-time is continuous or not. It is, however, interesting to observe that an initial apparatus of events suffices to generate a continuous space-time of points, by means of the relations of co-punctuality and logical inclusion. The further development of our geometry, so as to include surfaces, volumes, and four-dimensional regions, obviously presents no difficulty in principle, and I do not propose to enlarge upon it. I will merely observe that it is possible to extend the method by which we have defined points and lines so as to obtain something which we may call surfaces and regions, though not quite in the usual sense. Probably various ways of doing this are possible; the one that I suggest is the following. A class of lines will be called "co-superficial" when any two intersect, but there is no point common to all the lines of the class. A "surface" is a co-superficial class of lines which cannot be augmented without ceasing to be co-superficial. A class of surfaces is "co-regional" when any two have a line in common, but no line is common to all the surfaces of the class. A "region" is a co-regional class of surfaces which cannot be augmented without ceasing to be co-regional. It is obvious that this method could be extended to any number of dimensions; also that it requires limitations and extensions. But it seems unnecessary to pursue the matter further, since it is plain that we have what is needed for the pre-co-ordinate geometry of space-time. Let us now compare our constructed space-time with the spatial manifolds of analysis situs. In the preceding chapter[Pg 312] we quoted Hausdorff's definition of a "topological" space, and we saw that, in order to prove the usual propositions about limits, it is necessary that the total number of neighbourhoods should be . Let us now define as a "neighbourhood" of a point any set of points each of which contains as a sub-class a certain finite co-punctual class of events which is a sub-class of . That is to say, if a is a co-punctual class of events each of which is a member of , the set of all the points of which a is a sub-class will be a neighbourhood of . With this definition of a "neighbourhood," it is obvious that our space has the four characteristics by which Hausdorff (loc. cit., p. 213) defines a topological space. In order to insure that our space shall also satisfy his second denumerative axiom (loc. cit., p. 263), it is necessary and sufficient to assume that the total number of events is . With this assumption, the theorems of analysis situs become applicable to our space-time manifold of points. It remains to say a word on the subject of dimensions. We have not so far said anything explicit on this subject, though our original introduction of co-punctuality as a five-term relation could only prove satisfactory in a four-dimensional manifold. The most suitable definition of dimensions from our point of view is that of Poincaré, which is inductive. He defines a space as one-dimensional if, given any two points , , there is an isolated set of points such that no connected part of -not- contains both and . And he defines a space as -dimensional if, given any two points , , there is an ()-dimensional set of points such that no connected part of -not- contains both and . Using this definition, or any other which is purely topological, we set up the axiom that our topological space-time is to be four-dimensional.[67] This completes the material required for the topological treatment of space-time. FOOTNOTES: [66] For a geometry based on "neighbourhood," see Hausdorff, Grundzüge der Mengenlehre (Leipzig, 1914), chaps, VII. and VIII. [67] For an account of the modern theory of dimensions, see Karl Menger, Bericht über die Dimensionstheorie, Jahresbericht der deutschen Mathematiker-Vereinigung, 35, pp. 113-150 (1926). [Pg 313] CHAPTER XXX CAUSAL LINES THE notion of causality has been greatly modified by the substitution of space-time for space and time. We may define causality in its broadest sense as embracing all laws which connect events at different times, or, to adapt our phraseology to modern needs, events the intervals between which are time-like. Now owing to the fact that the formula for is formally the same for time-like and for space-like intervals, there is no longer the difference that formerly existed between causal and geometrical relations. Geodesics are geometrical, but they are also the paths of material particles. It is hardly correct to say that a particle moves in a geodesic; it is more correct to say that a particle is a geodesic (though not all geodesics are particles). To say that a particle moves in a geodesic is to use language appropriate to the conception of a space which persists through time, involving the notion of a position which may be occupied either at one time or at another. We think, for example, that it is possible to move from to or from to ; but such a view is incompatible with the theory of space-time. According to that theory, every position of a body has a date, and it is impossible to occupy the same position at another date, since the date is one of the co-ordinates of the position. When we travel from to , the date is continually advancing; the return journey, having different dates, does not cover the same route. Thus geometry and causation become inextricably intertwined. Dr A. A. Robb has laid stress upon the fact that, when two events have a space-like interval, there can be no direct causal relation between them. This means that, given two such events and , if any inference is possible from the one to[Pg 314] the other, it must be by way of a common causal ancestor. Two men may see the sun at the same moment, so that the interval between their percepts is space-like; the inference that so-and-so is seeing the sun now arises from our knowledge of radiation, and requires that we should trace his percept and our own to a common ancestry in the sun. We may therefore distinguish time-like and space-like intervals by saying that the former occur where there is some direct causal relation, while the latter occur where both events are related to a common ancestor or a common descendant. And possibly the magnitude of the interval may be derivable from the magnitude of the causal relation. But if this is to be possible, it will be necessary to achieve considerable precision as to what we mean by causal relations. As we saw in Part II., perception as a source of knowledge concerning physical objects would be impossible if there were not, in the physical world, semi-independent causal chains, or causal lines as we may call them. The light which comes to us from a printed page retains the structure of the page; if it did not, reading would be impossible. The retention is only approximate; it ceases at a distance from the book. And it ceases within the eye if we have defective vision. But where there is such failure, perception ceases—or rather, it fades away as the failure to preserve structure increases. Thus it is essential to perception as a source of knowledge that there should be in the world causal series which are, within limits, independent of the rest of the world. Another point concerning causation emerges from the consideration of perception. A number of simultaneous percepts—e.g. the letters of a word which we read at a glance—are to be regarded as "co-punctual" in the sense of our two preceding chapters. Each of these percepts has its own causal antecedents, different from those of the other percepts. It is true that there may be mutual modification—e.g. a[Pg 315] colour looks different in the neighbourhood of another colour from what it looks against a dark background. But this is recognized as "modification," i.e. as effecting a change from a norm, which must remain within limits if perception is to be successful. Thus the percipient is the meeting-place of a number of more or less independent causal series—as many, at least, as there are distinguishable elements in his total momentary perceptual field. But although these lines have converged upon him more or less independently, the totality of his percepts now becomes a causal unit, as is seen in mnemic phenomena. Given a number of simultaneous