The Analysis of Matter

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 percepts, a percept very similar to one of them, occurring on a future occasion, recalls something similar to the others, or at least may do so; here the co-punctuality of the percepts is essential to the character of their total effect. In the physical world, the same sort of thing must be supposed to occur, though to a less striking degree. According to the theory of Chapter XXVIII., any event in the physical world occupies a finite region of space-time, whose finiteness consists in the fact that the said event is compresent with events which are not compresent with each other. On the analogy of mnemic phenomena, a group of co-punctual events may have effects which would have been impossible if the events had not been co-punctual. That is the reason why physics is compelled to resort to points in stating its causal laws. Until we have a complete group of co-punctual events, i.e. a point, we cannot be quite sure as to the effect which will follow from any one of the events; such knowledge as we can have will be more or less approximate. It is these two opposite laws, of approximately separable causal lines on the one hand, and interactions of co-punctual events on the other, which make the warp and woof of the world, both physical and mental. In this chapter, I want to attain more precision as to the separable causal lines. [Pg 316] The possibility of perception, as we have seen already, depends upon the occurrence in the physical world of processes which may be called "radiations," provided the word is used somewhat more widely than is customary. The processes commonly called radiations are, naturally, the most perfect examples. In these, when they are undisturbed, we have a condition of some kind which spreads outward from a centre, changing in an apparently continuous manner as it travels. Something may be met with on the way which alters the law of change, or even stops the radiation in some direction altogether; but in the absence of obstacles the process proceeds according to its own intrinsic laws. The public senses—sight, hearing, and smell—depend upon radiations, in a generalized sense in the case of smell. Bodily senses, including touch, are more analogous to electric currents in their manner of propagation: they travel along nerves, but not through air or empty space. The public senses, also, travel along nerves, but the disturbance in the nerves is a prolongation, with alterations, of a process in the world outside the percipient's body, which is not the case with the bodily senses. It is owing to the existence of radiations that we live in a common world, since this depends upon the fact that neighbouring percipients receive similar stimuli at about the same time. The physical account of radiations is, however, very different in different cases. In the case of smell, the emission theory is universally accepted: we smell a body because portions of it travel from it to the nose. In the case of sound, only a process, not actual matter, is transmitted, but the process is in matter. In the case of light, if we accept the undulatory theory, the process consists of a transverse vibration, which may be said to be in the æther if that brings comfort to the speaker, but is certainly not in ordinary matter. If we could accept the light-quantum theory, we should still suppose that there is some periodic process, such that the action during one period[Pg 317] is (Planck's constant); the light consists of (so to speak) atoms, each of which is such a process. There is a great difference of physical importance between these three cases of smell, sound, and light; the first is quite unimportant physically, the second a somewhat late development from more fundamental principles, the third a corner-stone of physical theory. In the ideal case of a radiation, a few observations should suffice to determine its centre, and then, its laws being known, we could infer the whole connected system of events which constitutes it, in so far as the events enter into physical laws. The case of light from a fixed star very nearly realizes the ideal. The places in the universe where the light encounters obstacles are very few, though unfortunately they include the places where we live. It is because this example of light in vacuo is so nearly perfect that we know as much as we do about astronomy. Radiation independent of matter, however, is only one form of causal process in the physical world. Apart from quantum changes, there are at least two others which are of great