THE subject of this chapter is one which has been treated with wonderful ingenuity by Dr Whitehead, to whom is due the whole conception of a method which arrives at "points" as systems of finitely-extended events. In advocating this method, it is not necessary to maintain that mathematical points are impossible as simple entities (or "particulars"); all that it is necessary to maintain is that we have no good ground for regarding them as such. What we know about points is that they are useful technically—so useful that we must seek an interpretation of the propositions in which, symbolically, they occur. But there is no ground for denying structure to a point; on the contrary, there are two grounds for assigning structure to a point. One is the familiar argument of Occam's razor: we can make structures having the mathematical properties of points, and to suppose that there are points in any other sense is an inference which is useless to science and not warranted by any principle, logical or scientific. The other argument is much more difficult to state, but the more one studies logical construction the more weight one feels inclined to attach to it. It rests upon a maxim which might be enunciated as a supplement to Occam's razor: "What is logically convenient is likely to be artificial." To me personally, the first example of this maxim was the definition of real numbers. Mathematicians found it convenient to suppose that all series of rationals have limits, while nevertheless some do not have rational limits. They therefore postulated irrational limits, supposed to be homogeneous with the rationals. Although the method of Dedekind cuts was familiar, nobody thought of saying: An irrational is a Dedekind cut, or at least its inferior portion. Yet this definition solves all difficulties. We have now first ratios (which cannot be irrational), then segments of the series of ratios. Segments which have a limit are rational, segments which have no limit are irrational. The square root of 2 is the class of ratios whose square is less than 2. Segments of the series of ratios are real numbers the series of real numbers has both Dedekindian and Cantorian continuity. Thus it is mathematically convenient; but its logical structure is more complex than that of the series of ratios. The logical analysis of mathematics affords many examples of this procedure, such as the construction of "ideal" points, lines, and planes alluded to in Chapter XX.

It will be seen that the phrase "what is logically convenient is artificial" does not express what is meant with as much precision as is to be desired. What we mean is this: Given a set of terms having properties which suggest certain general mathematical (or logical) properties, but are subject to exceptions in regard to these properties, it is a mistake to postulate other terms, logically homogeneous with the original set, and such as to remove the exceptions; the proper procedure is to look for logical structures composed of the original terms, and such that these structures always have the mathematical properties in question. It will be found that, where the assumption of such properties has proved fruitful, this procedure is usually possible.

Starting from events, there are many ways of reaching points. One is the method adopted by Dr Whitehead, in which we consider "enclosure-series." Speaking roughly, we may say that this method defines a point as all the volumes which contain the point. (The niceties of the method are required to prevent this definition from being circular; also to distinguish a set of volumes having only a point in common from such as have a line or surface in common.) As a piece of logic, this method is faultless. But as a method which aims at starting with the actual constituents of the world it seems to me to have certain defects. Dr Whitehead assumes that every event encloses and is enclosed by other events. There is, therefore, for him, no lower limit or minimum, and no upper limit or maximum, to the size of events. Each of these assumptions demands consideration.

Let us begin with the absence of a lower limit or minimum. Here we are confronted with a question of fact, which might conceivably be decided against Dr Whitehead, but could not conceivably be decided in his favour. The events which we can perceive all have a certain duration, i.e. they are simultaneous with events which are not simultaneous with each other. Not only are they all, in this sense, finite, but they are all above an assignable limit. I do not know what is the shortest perceptible event, but this is the sort of question which a psychological laboratory could answer. We have not, therefore, direct empirical evidence that there is no minimum to events. Nor can we have indirect empirical evidence, since a process which proceeds by very small finite differences is sensibly indistinguishable from a continuous process, as the cinema shows. Per contra, there might be empirical evidence, as in the quantum theory, that events could not have less than a certain minimum spatio-temporal extent. Dr Whitehead's assumption, therefore, seems rash. At the same time, there is a confusion to be avoided: space-time may be continuous even if there is a lower limit to events. Suppose every elementary event filled a four-dimensional cube, e.g. a cubic centimetre lasting for the time that light takes to travel a centimetre; and suppose, conversely, that every such four-dimensional cube was occupied by an event. The space-time of such a world would be continuous, given suitable axioms, although events had a minimum. And, conversely, the absence of a minimum to events does not insure spatio-temporal continuity. The two questions are thus wholly distinct.

I conclude that there is at present no means of knowing whether events have a minimum or not; that there never can be conclusive evidence against their having a minimum; but that conceivably evidence may hereafter be found in favour of a minimum. It remains to consider the question of a maximum.

On the question of a maximum to events, the arguments are rather logical than empirical. In a certain sense, any series of events may be called one event; the Battle of Waterloo, for instance, may count as a single occurrence. But in a complex event of this sort, there are parts which have spatio-temporal and causal relations to each other; no single entity devoid of physical structure persists throughout the whole period. I mean by this that anything simultaneous with everything that happened during the Battle of Waterloo is a complex of parts not all simultaneous with each other. Whether we are to call such a complex an "event" or not is merely a question of words. But if our object is to exhibit the structure of the physical world, it is clear that we 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, 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:

(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."

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]

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) 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, 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 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 and and co-punctual with , , , , , , then must be later in the -series than . If this process comes to an end with , then , , , , , , ... together form a point. If it does not come to an end with any finite , it may happen that no outside the series (, , ... ,...) is co-punctual with , , , and all the 's; in that case, , , , and these 's form a point. But if there are 's other than the 's and co-punctual with all of them, let be the first of them. Then is later in the -series than any of the finite 's. We proceed in this way as long as possible, using two principles: (1) given a series of 's ending with , let be the first in the -series after and co-punctual with the group of all the previous 's; (2) given a series of 's having no last term, take as the next the first in the -series which is after all the 's hitherto selected and co-punctual with all of them. If, at any stage, there is no such , the 's already selected form a point. Now this process must end sooner or later; for the 's (other than ) form an ascending series selected from , and therefore, sooner or later, there will be no 's later than all the 's previously selected. At this stage, if not before, , , , and the 's already selected will form a point. Hence if all events can be well ordered, every event is a member of at least one point, provided every event is a member of a co-punctual quintet. The proof still holds if we only assume that all events co-punctual with a given quintet can be well ordered.

Given any class of events , let be the class of those events which are co-punctual with a. Then by definition a is a point if . The necessary and sufficient condition that all the members of a should have a point in common is that a should be contained in . This condition is necessary, for, if is a point and is contained in , it follows that is contained in , and that , so that is contained in . The proof that the condition is sufficient is longer; it is as follows.

If , is a point. If not, let denote the part of which is outside . Using again the -series of all events, put and so on, as long as possible. If , precedes in the -order. Hence, as before, there must come a stage when no fresh 's can be constructed. If is the class consisting of a together with all the 's yielded by the method, is a point. For (1) all the quintets in are co-punctual, by the construction; (2) a term co-punctual with all the quartets of cannot be later than all the 's, because if there were such a term we could construct more 's; (3) such a term cannot be earlier than some member of because, if it were, it would have been chosen as the of that stage in the construction; hence no event outside is co-punctual with every quartet of . Hence is a point.

To say that a collection of events have a point in common is to say that the collection is part (or the whole) of the class which is the point. Conversely, a collection of events may contain a sub-class which is a point; the necessary and sufficient condition for this is that should be contained in , where is the collection in question. The proof proceeds exactly as before, if we now make mean the part of which is not contained in .

A group of events a is "co-punctual" if is contained in , and a "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual.

A few purely logical properties of points may be noted. Given any two classes and , if is contained in , then is contained in . Hence if and are points and is contained in , and are identical; for in that case and are respectively identical with and , and therefore if is contained in , is contained in , so that and are identical.

Every co-punctual group of events contains at least one point. This has already been proved, since to say that a is a co-punctual group is to say that a is contained in .

It may be taken that, in general, there are a number of points of which any given event is a member. Such a set of points will fill a "region," but not every region will be the set of points to which some one event belongs. This topic, however, cannot be dealt with until we have discussed space-time order.