The Analysis of Matter

begin with. We now define " is between and " as meaning: ", , are points such that is not null and is a proper part of ." An equivalent definition is: ", , are points such that is not null, and is contained in , but is not contained in ." By the help of suitable axioms, "between," so defined, can be made to give rise to the spatio-temporal order presupposed in assigning co-ordinates in the general theory of relativity. What the definition says, in geometrical language, is that every event which contains both and contains , but not every event which contains both and contains . We must not imagine that all the points between two others lie on one line; each lies on some short route joining the end-points, a "short" route being one composed wholly of points between the end-points; but none lies on all short routes. Before developing the formal consequences of this definition, it may be as well to consider its geometrical import. In the accompanying figure, will be between and if there are events which contain all three, but there are none which contain and without containing . (I represent events by areas.) Now if events can often be of irregular shapes[Pg 306] such as that of the shaded area in the figure, it would seem that one event is not likely ever to be between two others according to the definition. I shall therefore assume that we may picture events as free from re-entrant angles and similar oddities. I imagine them as all oval; but formally it would do just as well if they were all four-dimensional cubes, and it would not matter whether they were large or small, provided they did not differ too much, and were all above a certain minimum. These pictorial requisites are rather for the importance of the theory to be developed than for its truth. In the preceding chapter, we assumed that events are such as to be all spheres according to one possible metric. Formally, we might equally well have assumed that there is a metric in which they are all cubes. Some assumption of this kind, as we saw, is necessary for the success of our definition of points. The other assumptions needed for its truth will be explicitly stated as they are introduced. The assumptions introduced so far in this chapter and its predecessor are: (1) Compresence is symmetrical. (2) Defining "events" as the field of compresence, every event is compresent with itself. (3) Events can be well ordered; or at least those compresent with a given event can be. (4) Any two events have a relation which is a finite power of compresence. (This is required for mapping space-time into zones.) In other words, the ancestral relation derived from compresence is connected. We will now define a set of points as "collinear" if every pair of the set are connected, and every triad , , are such that either is contained in , or is contained in . We will define a set of points as a "line" if (1) it is collinear, (2) it is not contained in any larger collinear group with the same extremities. It will be seen that this definition is analogous to that of points. We may define a set of events[Pg 307] as "co-punctual" when every quintet of the set are co-punctual; and we can then define a set of events as a "point" when (1) it is co-punctual, (2) it is not contained in any larger co-punctual group. This way of stating our previous definition of "points" brings out the analogy. The "lines" that we are defining are not to be supposed "straight"; straightness is a notion wholly foreign to the geometry we are developing. Perhaps it might be better to call them "routes"; but there is no harm in calling them "lines" provided we remember that they are not supposed to be straight. For the present, we shall not be concerned with lines, but only with collinear groups of points. Let us define a set of points as "-collinear" if (1) every pair of the set is connected; (2) given any two, , , either is between or , or is between and . We shall want such axioms as will enable us to show that such a set of points is collinear, not merely -collinear, and that their order is independent of . It is obvious that, if we put before whenever is between and , we obtain a serial order of any set of points which is -collinear. But to insure that the order shall be independent of we require the following three axioms: (1) If , , , are points, and is contained in , and is contained in , and and are distinct, then is not contained in . (2) If is contained in , and is contained in , than is contained in the sum of and . (It follows at once that is contained in .) (3) If is contained in , and is contained in , then is contained in the sum of and . (It follows at once that is contained in .) The practical effects of these three axioms are: [Pg 308] (1) If and are between and , and is between and , then is not between and . (2) If is between and , and is between and , then and are between and . (3) If is between and , and is between and , then is between and . From these axioms we can deduce that a set of points which is -collinear is collinear. Also that, given a set of -collinear points, if is one of them, the points of the set which are beyond from a are -collinear, and retain the same order when arranged with reference to as they had when arranged with reference to . Also that, if is one of a set of -collinear points, those of the set which are between and are -collinear, and have, when arranged with reference to , the converse order to that which they had when arranged with reference to . These propositions show that we have a satisfactory definition of order among the points of a collinear set. The above axioms are logically adequate, but regarded as asserting physical truths about events they may perhaps be regarded as more or less doubtful. We have to remember that our lines are not straight, and may therefore return into themselves. Routes with very great curvature are, however, excluded by our definition of collinearity. Consider, e.g., such a route as that in the accompanying figure. We may suppose that , , , are all connected, but and will not be between and according to the definition, because obviously an event may contain and without containing and . Thus if we wish to regard the above route from to as, in some sense, a line, it will have to be in an extended sense, namely, that it can be divided into a number of small finite parts, each of which is a line. And a set of points may be regarded as collinear in an extended sense if it is capable[Pg 309] of a serial order such that any sufficiently small consecutive stretch of the series is a collinear set—provided that such stretch must contain not less than four points. We can now prove, by the help of one further axiom, that any progression of collinear points all lying between two points and must have a limit. Let our set of points be all lying on a line between and , in an order from towards . Let be the sum of all the points 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