The Concept of Nature — the Tarner Lectures, 1920

The required character of the abstractive sets which form event-particles would be secured if we could define them as hiving the property of being covered by any abstractive set which they cover. For then any other abstractive set which an abstractive set of an eventparticle covered, would be equal to it, and wouldp. 87 therefore be a member of the same event-particle. Accordingly an event-particle could cover no other abstractive element. This is the definition which I originally proposed at a congress in Paris in 1914^ There is however a difficulty involved in this definition if adopted without some further addition, and I am now not satisfied vdth the way in which I attempted to get over that difficulty in the paper referred to.

The difficulty is this : When event-particles have once been defined it is easy to define the aggregate of eventparticles forming the boundary of an event ; and thence to define the point-contact at their boundaries possible for a pair of events of which one is part of the other. We can then conceive all the intricacies of tangency. In particular we can conceive an abstractive set of which all the members have point-contact at the same event-particle. It is then easy to prove that there will be no abstractive set with the property of being covered by every abstractive set which it covers. I state this difficulty at some length because its existence guides the development of our line of argument. We have got to annex some condition to the root property of being covered by any abstractive set which it covers. When we look into this question of suitable conditions we find that in addition to event-particles all the other relevant spatial and spatio-temporal abstractive elements can be defined in the same way by suitably varying the conditions. Accordingly we proceed in a general way suitable for employment beyond event-particles.

Let a be the name of any condition which some abstractive sets fulfil. I say that an abstractive set is

1 Cf. 'La Thdorie Relationniste de I'Espace,' Rev. de MStaphysique et de Morale, vol. xxiii, 1916.

p. 88

* a -prime' when it has the two properties, (i) that it satisfies the condition o and (ii) that it is covered by every abstractive set which both is covered by it and satisfies the condition a .

In other words you cannot get any abstractive set satisfying the condition a which exhibits intrinsic character more simple than that of a a -prime.

There are also the correlative abstractive sets which I call the sets of a-antiprimes. An abstractive set is a cr -antiprime when it has the two properties, (i) that it satisfies the condition a and (ii) that it covers every abstractive set which both covers it and satisfies the condition a. In other words you cannot get any abstractive set satisfying the condition a which exhibits an intrinsic character more complex than that of a CT -antiprime.

The intrinsic character of a o-prime has a certain minimum of fullness among those abstractive sets which are subject to the condition of satisfying a\ whereas the intrinsic character of a a -antiprime has a corresponding maximum of fullness, and includes all it can in the circumstances.

Let us first consider what help the notion of anti-primes could give us in the definition of moments which we gave in the last lecture. Let the condition «T be the property of being a class whose members are all durations. An abstractive set which satisfies this condition is thus an abstractive set composed wholly of durations. It is convenient then to define a moment as the group of abstractive sets which are equal to some <T-antiprime, where the condition a has this special meaning. It will be found on consideration (i) that each abstractive set forming a moment is a cr-antiprime,p. 89 where a has this special meaning, and (ii) that we have excluded from membership of moments abstractive sets of durations which all have one common boundary, either the initial boundary or the final boundary. We thus exclude special cases which are apt to confuse general reasoning. The new definition of a moment, which supersedes our previous definition, is (by the aid of the notion of antiprimes) the more precisely drawn of the two, and the more useful.

The particular condition which V stood for in the definition of moments included something additional to anything which can be derived from the bare notion of extension. A duration exhibits for thought a totality. The notion of totality is something beyond that of extension, though the two are interwoven in the notion of a duration.

In the same way the particular condition ' a ' required for the definition of an event -particle must be looked for beyond the mere notion of extension. The same remark is also true of the particular conditions requisite for the other spatial elements. This additional notion is obtained by distinguishing between the notion of 'position ' and the notion of convergence to an ideal zero of extension as exhibited by an abstractive set of events.

In order to understand this distinction consider a point of the instantaneous space which we conceive as apparent to us in an almost instantaneous glance. This point is an event-particle. It has two aspects. In one aspect it is there, where it is. This is its position in the space. In another aspect it is got at by ignoring the circumambient space, and by concentrating attention on the smaller and smaller set of events which approximate to it. This is its extrinsic character. Thus a point hasp. 90 three characters, namely, its position in the whole instantaneous space, its extrinsic character, and its intrinsic character. The same is true of any other spatial element. For example an instantaneous volume in instantaneous space has three characters, namely, its position, its extrinsic character as a group of abstractive sets, and its intrinsic character which is the limit of natural properties which is indicated by any one of these abstractive sets.

Before we can talk about position in instantaneous space, we must evidently be quite clear as to what we mean by instantaneous space in itself. Instantaneous space must be looked for as a character of a moment. For a moment is all nature at an instant. It cannot be the intrinsic character of the moment. For the intrinsic character tells us the limiting character of nature in space at that instant. Instantaneous space must be an assemblage of abstractive elements considered in their mutual relations. Thus an instantaneous space is the assemblage of abstractive elements covered by some one moment, and it is the instantaneous space of that moment.

We have now to ask what character we have found in nature which is capable of according to the elements of an instantaneous space different qualities of position. This question at once brings us to the intersection of moments, which is a topic not as yet considered in these lectures.

The locus of intersection of two moments is the assemblage of abstractive elements covered by both of them. Now two moments of the same temporal series cannot intersect. Two moments respectively of diiferent families necessarily intersect. Accordingly in the in-

p. 91