The Analysis of Matter

must distinguish objects having physical structure from such as are only component parts of such structures. It is therefore convenient to have a word for the latter. The word I shall use is "event." But I shall not go so far as to say that an "event" must have no structure. I shall assume only that any structure which it may have is irrelevant both to physics and to psychology; in other words, that its parts, if any, do not have scientifically distinguishable relations to other objects. When the word "event" is used in this sense, it is plain that,[Pg 294] so far as our experience goes, no event lasts for more than a few seconds at most. There is no a priori reason why this should be the case; it is merely an empirical fact. But I think a phraseology which obscures it can only lead to confusion. For the above reasons, I am unable to accept Dr Whitehead's construction of points by means of enclosure-series as an adequate solution of the problem which it is designed to solve. This problem is: to discover structures having certain geometrical properties, and composed of the raw material of the physical world. There is another method, which may be called that of "partial overlapping." In my Knowledge of the External World, I applied this method to the definition of instants. It is easy to see that it is adequate for this purpose in psychology, where we have a one-dimensional time-order which remains definite in spite of relativity. But in physics it is the "point-instant" that has to be defined, i.e. a completely definite position in space-time, not merely in space or merely in time. Here the method is only applicable with suitable modifications. However, the method must first be explained as applied to the one-dimensional psychological time-series. We assume that two events may have a relation which I will call "compresence," which means, practically, that they overlap in space-time. Take, for instance, notes played by different instruments in orchestral music: if one is heard beginning before the other has ceased to be heard, the auditory percepts of the hearer have "compresence." If a group of events in one biography are all compresent with each other, there will be some place in space-time which is occupied by all of them. This place will be a "point" if there is no event outside the group which is compresent with all of them. We may therefore define a "point-instant," or simply a "point," in one biography, as a group of events having the following two properties: [Pg 295] (1) Any two members of the group are compresent; (2) No event outside the group is compresent with every member of the group. When we pass beyond one dimension, this method is no longer applicable. Take, for example, the three circles in the accompanying figure: each overlaps with the other two, but there is no region common to all three. If we try to remedy this (as I believe we can) by starting, in two dimensions, with a relation of three events, which is to hold when all three have a region in common, we are still met by difficulties. The three circles , , have a region in common, and the shaded area has a region in common with and , also with and , and also with and , yet , , and have no region in common. Therefore if events may have queer shapes such as , our new three-term relation will still not enable us to define a "point." [Pg 296] Since the problem with which we are concerned belongs to analysis situs, in which we are occupied only with such properties of figures as are unaffected by continuous deformation, we cannot simply declare in advance that no events are to have odd shapes. But before attempting to deal with this difficulty, it will be as well to consider certain points in analysis situs, which will show us what are the requisites of a solution of our problem. In analysis situs we start with two conceptions, that of a point, and that of "neighbourhoods of a given point"—the latter being collections of points. Certain definitions obtained in this way will be useful. The following definitions are due to Leopold Vietoris.[60] If is a set of points, a point is called a "Häufungspunkt" of if in every neighbourhood of there is a point other than . Two collections of points "touch" each other in a point if belongs to one collection and is a "Häufungspunkt" of the other. A set of points is "continuous from to " if it contains and , and any two parts of it whose sum is , of which one contains and the other , touch each other (in at least one point). A set of points is a "Linienstück" from to if it, but none of its proper parts, is continuous from to . Hausdorff[61] has defined a "metrical" space and a "topological" space in the following terms. A "metrical" space is a manifold such that with any two points , is associated a real not-negative number having the following three properties: (a) ; (b) is only zero when and are identical; (c) is greater than or equal to .[62] A "topological" space is a manifold whose elements are associated with sub-classes of the manifold such that: (A) To every corresponds at least one , and every contains ; (B) If , are both neighbourhoods of there is a neighbourhood of , say which is contained in the common part of and ; (C) If y is a member of , there is a neighbourhood of which is contained in ; (D) Given any two distinct points, there is a neighbourhood of the one and there is a neighbourhood of the other such that the two have no common point.[63] [Pg 297] In order to be able to apply the usual methods of limits to a topological space, Hausdorff has need of an "Abzählbarkeitsaxiom," or "denumerative axiom." He gives two such axioms (p. 263), of which the first is the weaker, and is for some purposes insufficient. The first states that the number of neighbourhoods of a given point is never greater than ; the second states that the total number of neighbourhoods of all points is together . This second axiom suffices for all the usual kinds of argument, without the introduction of any metrical ideas. P. Urysohn[64] has shown that every topological space which satisfies Hausdorff's second denumerative axiom and has one further property (which he calls "normality"[65]) is metricizable. These are the main points from analysis situs that are relevant to the solution of our problem. For the present, we are not concerned with metrical properties, but only with such as belong to "topological" spaces. In virtue of Urysohn's theorem, it will be possible to introduce a metric if we can construct the right sort of topological space. But when one metric is possible, an infinite number are possible. The metric which is actually introduced in theory of relativity is introduced for empirical reasons; it uses a quantitative relation which might be called degree of causal proximity. The existence of this relation is not implied by anything with which we are at present concerned. Moreover, the metrical manifold which we require in physics is not a "metrical space" according to Hausdorff's definition given above, since interval in relativity does not possess the properties (b) and (c)[Pg 298] which distance possesses in Hausdorff's definition. However, so far as topological considerations are concerned, we may, without appreciable inaccuracy, assign to small regions the topological properties which belong to a small region of Euclidean space lasting for a short time, i.e. to a continuous series of small regions of Euclidean space all geometrically indistinguishable. In analysis situs, both points and neighbourhoods are given. We, on the other hand, wish to define our points in terms of "events" where "events" will have a one-one correspondence with certain neighbourhoods. We want our "events" to correspond with neighbourhoods which are above a certain minimum and below a certain maximum when, at a later stage, the empirical metric is introduced. We have to assign to our events such properties as will enable us to define the points of a topological space as classes of events, and the neighbourhoods of the points as classes of points. But we have to remember that we do not want to construct merely a topological space: what we want to construct is the four-dimensional space-time of the general theory of relativity. The following illustration will serve to introduce the problem. Consider a three-dimensional Euclidean numerical space, i.e. the manifold of all ordered triads of real numbers (, , ), with the usual definition of distance. Consider, in this space, all the spheres having a given radius and having centres whose co-ordinates are rational. The number of such spheres is . Let us define a group of these spheres as "co-punctual" if it is such that every four chosen out of the group have a common region; and let us define a co-punctual group as "punctual" if it cannot be enlarged without ceasing to be co-punctual. Then there is a one-one correspondence between the original points of our space and the punctual groups of spheres. Consequently the punctual groups of spheres form a Euclidean space. If the spheres are all distorted in any continuous way,[Pg 299] they will still enable us to construct punctual groups in the same way, and the manifold of punctual groups will still have all the topological properties which are possessed by a three-dimensional Euclidean space. Therefore if we are to use this method of constructing points out of "events," we shall have to assume that, in the resulting space, there is a possible metric according to which the points of which a given event is a member always form a spherical volume. Although this is expressed in metrical language, it is in reality a topological property, since it is unaffected by continuous deformation. It must be possible to express it in non-metrical language, though I must confess that I lack the necessary skill. I propose, therefore, to regard events as occupying regions of space-time which, in some possible metric, are spheres so far as their space-dimensions are concerned, and between a certain maximum and a certain minimum so far as their time-dimension is concerned. The region "occupied" by an event is the class of points of which it is a member. As the fundamental relation in the construction of points, we take a five-term relation of "co-punctuality," which holds between five events when there is a region common to all of them. A group of five or more events is called "co-punctual" when every quintet chosen out of the group has the relation of co-punctuality. A "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. In order to demonstrate the existence of points so defined, it is sufficient to assume that all events (or at least all events co-punctual with a given co-punctual quintet) can be well ordered. If Zermelo's axiom is true, this must be the case; if not, it may involve some limitation as to the number of events. I have been led by the arguments, first of Dr H. M. Sheffer, and then of Mr F. P. Ramsey, to the view that Zermelo's axiom is true; I am therefore less reluctant than I[Pg 300] should have been formerly to assume that events can be well ordered. To prove that every event is a member of at least one point, we proceed as follows—assuming that there are co-punctual quintets. Let be a well-ordered series whose field consists of all events; put Let , , , , be a co-punctual quintet. If is the only event co-punctual , , , , then the class whose only members are , , , , is a point according to the definition. If, on the other hand, there are 's other than which are co-punctual with , , , , , let be the first of them. If no other than and is co-punctual with , , , , and , then , , , , and form a point. Otherwise, let be the first other than