The Concept of Nature — the Tarner Lectures, 1920

The point which I want to make now is that being the situation of a well-marked object is not an inherent necessity for an event. Wherever and whenever something is going on, there is an event. Furthermore ' wherever and whenever ' in themselves presuppose an event, for space and time in themselves are abstractions from events. It is therefore a consequence of this doctrine that something is always going on everywhere, even in so-called empty space. This conclusion is in accord with modern physical science which presupposes the play of an electromagnetic field throughout space and time. This doctrine of science has been thrown into the materialistic form of an all-pervading ether. But the ether is evidently a mere idle concept — in the phraseology which Bacon applied to the doctrine of final causes, it is a barren virgin. Nothing is deduced from it; and the ether merely subserves the purpose of satisfying the demands of the materialistic theory. The important concept is that of the shifting facts of the fields of force. This is the concept of an ether of events which should be substituted for that of a material ether.

It requires no illustration to assure you that an event is a complex fact, and the relations between two events form an almost impenetrable maze. The clue discovered by the common sense of mankind and systematicallyp. 79 utilised in science is what I have elsew^here ^ called the law^ of convergence to simplicity by diminution of extent.

If A and B are two events, and A' is part of A and B' is part of B, then in many respects the relations between the parts A' and B' will be simpler than the relations between A and B. This is the principle which presides over all attempts at exact observation.

The first outcome of the systematic use of this law has been the formulation of the abstract concepts of Time and Space. In the previous lecture I sketched how the principle was applied to obtain the time-series. I now proceed to consider how the spatial entities are obtained by the same method. The systematic procedure is identical in principle in both cases, and I have called the general t}'pe of procedure the ' method of extensive abstraction.'

You will remember that in my last lecture I defined the concept of an abstractive set of durations. This definition can be extended so as to apply to any events, limited events as well as durations. The only change that is required is the substitution of the word ' event ' for the word ' duration.' Accordingly an abstractive set of events is any set of events which possesses the two properties, (i) of any two members of the set one contains the other as a part, and (ii) there is no event which is a common part of every member of the set. Such a set, as you will remember, has the properties of the Chinese toy which is a nest of boxes, one within the other, with the difference that the toy has a smallest box, while the abstractive class has neither a smallest

^ Cf. Organisation of Thought, pp. 146 et seq. Williams and Norgate, 1917.

8o THE CONCEPT Ut JNAi uku l^«.

event nor does it converge to a limiting event which is ftot a member of the set.

Thus, so far as the abstractive sets of events are concerned, an abstractive set converges to nothing. There is the set with its members grovidng indefinitely smaller and smaller as we proceed in thought towards the smaller end of the series; but there is no absolute minimum of any sort which is finally reached. In fact the set is just itself and indicates nothing else in the way of events, except itself. But each event has an intrinsic character in the way of being a situation of objects and of having parts which are situations of objects and — to state the matter more generally — in the way of being a field of the life of nature. This character can be defined by quantitative expressions expressing relations between various quantities intrinsic to the event or between such quantities and other quantities intrinsic to other events. In the case of events of considerable spatio-temporal extension this set of quantitative expressions is of bewildering complexity. If e be an event, let us denote by q (e) the set of quantitative expressions defining its character including its connexions with the rest of nature. Let e^, €2, e^, etc. be an abstractive set, the members being so arranged that each member such as e^ extends over all the succeeding members such as e^+i, e^+^y and so on. Then corresponding to the series there is the series

Call the series of events 5 and the series of quantitative expressions q (s). The series 5 has no last term and

IV] METHOD OF EXTENSIVE ABSTRACTION 8i

no events w^hich are contained in every member of the series. Accordingly the series of events converges to nothing. It is just itself. Also the series q {$) has no last term. But the sets of homologous quantities running through the various terms of the series do converge to definite limits. For example if Q^^ be a quantitative measurement found in q {e^, and Q^ the homologue to Q^ to be found in q {e^, and Q^ the homologue to Q^ and Q^ to be found in q {e^, and so on, then the series yi> ^2' >CZy '"■> icny >^n+li •••> though it has no last term, does in general converge to a definite limit. Accordingly there is a class of limits l{s) w^hich is the class of the limits of those members of g'(e„) v^hich have homologues throughout the series q{s) as n indefinitely increases. We can represent this statement diagrammatically by using an arrow (-*) to mean 'converges to.' Then ei, e^, ^3, ..., Bn, e^+i, ... ■* nothing, and

The mutual relations betv^^een the limits in the set l{s), and also betvsreen these limits and the limits in other sets l{s'), lis"), ..., v^hich arise from other abstractive sets s', s", etc., have a pecuHar simplicity.

Thus the set 5 does indicate an ideal simplicity of natural relations, though this simplicity is not the character of any actual event in s. We can make an approximation to such a simplicity which, as estimated numerically, is as close as we like by considering an event which is far enough down the series towards the small end. It will be noted that it is the infinite series.

p. 82

as it stretches away in unending succession towards the small end, which is of importance. The arbitrarily large event with which the series starts has no importance at all. We can arbitrarily exclude any set of events, at 'the big end of an abstractive set without the loss of any important property to the set as thus modified.

I call the limiting character of natural relations which is indicated by an abstractive set, the ' intrinsic character ' of the set; also the properties, connected with the relation of whole and part as concerning its members, by which an abstractive set is defined together form what I call its 'extrinsic character.' The fact that the extrinsic character of an abstractive set determines a definite intrinsic character is the reason of the importance of the precise concepts of space and time. This emergence of a definite intrinsic character from an abstractive set is the precise meaning of the law of convergence.