§ 7. (a) The Puzzle of Continuity. Continuity is, strictly speaking, a property of certain series, and may be defined for purposes of reference much as follows. A series is continuous when any term divides the whole series unambiguously into two mutually exclusive parts which between them comprise all the terms of the series, and when every term which so divides the series is itself a term of the series. From this second condition it obviously follows that a number of intermediate terms can always be inserted between any two terms whatever of a continuous series; no term of the series has a next term. This is the peculiarity of the continuous with which we shall be specially concerned. Thus the series of points on a straight line is continuous because (1) any point P on the line divides it into two collections of points in such a way that every point of the one is to the left of every point of the other, and every point of the second to the right of every point of the former; and (2) every point which divides the line in this way is a point on the line. Again, the whole series of real numbers is continuous for the same reason. Every member of the number-series divides it into two classes, so that every number of one is less than every number of the other, and every number which thus divides the series is itself a term of the number-series.
But the series of rational real numbers is not continuous, because it can be divided into mutually exclusive classes by terms which are not themselves members of the series. (E.g. √2 is not a member of the series of rational numbers, but we can exhaustively divide all rational numbers into the two mutually exclusive classes, rational numbers less than √2 and rational numbers not less than √2.)[[105]] From the continuity of the series of real numbers it follows that any other series which corresponds point for point with the terms of the number-series will be continuous. Now one such series is that of the successive parts of time. Every moment of time divides the whole series of moments into two mutually exclusive classes, the moments before itself and the moments which are not before itself. And whatever thus divides the time-series is itself a moment in that series. Hence from the continuity of the time-series it follows that any puzzles created by this property of continuousness will apply to the case of Causation. In what follows I shall not discuss the general problem of the continuous, a problem which requires special mathematical equipment for its efficient handling, but shall confine myself to the difficulties introduced by continuity into the scientific concept of causal relation.
We may conveniently attack the problem by taking it up in the form in which Hume bequeathed it to modern science. As any careful reader of Hume must perceive, Hume’s whole doctrine of Causation is based on the assumption that the causal process is not continuous. Experience is supposed by him to come to us not in an unbroken stream, but in isolated separate pieces which we subsequently proceed to link together artificially by the notion of Causation. We are supposed to begin by observing the sequence of an event B on a previous distinct event A, and the problem of Causation thus becomes that of discovering the nature of the link by which the originally distinct A and B are connected in our scientific thought. In more technical language, Hume thought of the series of events as one in which every member has a next term, and this way of conceiving it has coloured the whole subsequent treatment of Causation by the inductive logicians who have commonly got their metaphysical doctrines from Hume.
Now, recent Psychology, in deserting the old notion of the atomic sensation for that of the “stream of consciousness,” has completely destroyed the supposed empirical foundation for this Humian theory of the discontinuity of the course of events. The real problem for the inductive logician we can now see to be not to discover the link by which an originally separate A and B have got joined together in thought, but to find the source of the distinction we habitually draw within what comes to us as one continuous process between an earlier stage A which we call cause, and a later stage B which we call effect. We are not, however, concerned here with the psychological weakness of Hume’s doctrine, but with the logical difficulty to which it gives rise.
We may state the difficulty thus: (1) Causation cannot possibly be thought of as discontinuous, i.e. as the sequence of one distinct event upon an assemblage of other events without gross contradiction. To think of it as discontinuous, we must conceive the cause A to exist first in its completeness, and then to be suddenly followed by the effect B. (That the cause A consists of a number of conditions, a, b, c ... which themselves come into existence successively, and that A is not there until the last of these conditions has been realised, makes no difference to the principle.) Now this seems to be what is actually implied by the language of those inductive logicians who insist that in all Causation the cause must precede the effect. But what can such precedence mean? It can only mean that after the complete realisation of the conditions included in the cause A, there must intervene a space of empty time before the effect B enters on the scene. However brief and “momentary” you take this gap in the stream of events to be, the gap must be there if your language about the cause as being before the effect is to have any meaning. For if there is no such gap, and the entrance of B is simultaneous with the complete realisation of its conditions A, it is no longer true to say that the cause A is before the effect B. A does not exist as A until a, b, c ... are all present, and as soon as they are present B is present too. And thus the relation between A and B is not that of the sequence of a later event on an earlier. They are actually together.
In fact, the doctrine that the cause precedes the effect rests upon the notion that the time-series is one in which each member has a next term. And this seems inconceivable. For not only can you subdivide any finite time, however small, into two mutually exclusive parts, but the point at which the division is effected is itself a moment in the time-series lying between the beginning and the end of the original interval. Time therefore must be continuous, and if causation is not equally continuous, we must suppose that gaps of empty time are what separate the first event, the cause, from the subsequent event, the effect. Yet if this could be regarded as a defensible doctrine on other grounds, it would then follow that the assemblage of events A is not the totality of conditions requisite for the occurrence of B. The “totality of conditions,” i.e. the cause as previously defined, would be the events A plus a certain lapse of empty time.[[106]] And so the cause would once more turn out not to precede the effect, or we should have to suppose the end of the interval of empty time included in it as separated from the beginning of B by a second lapse, and so on indefinitely.
(2) These difficulties, in a more or less clearly apprehended form, have led many recent writers on inductive Logic to modify the definition which was still satisfactory to Mill. Cause and effect, we are now told, are not distinct events, but earlier and later stages in a continuous process. The real business of science is not to discover “laws of connection” between distinct events or “phenomena,” but to invent general mathematical formulae by the aid of which we may trace the course of continuous processes. The discovery of causes, from this point of view, is reduced to the construction of formulæ which exhibit some quantity as a function of a time-variable. Fully worked out, this view of the nature of experimental science leads to the so-called “descriptive” ideal of scientific explanation, advocated by such eminent thinkers as Kirchhoff, Mach, and Ostwald among physicists, and, with various modifications, Avenarius, Münsterberg, Royce, and James Ward among recent philosophers. According to this doctrine, the ultimate ideal of science, or at any rate of physical science, is simply the description of the course of events by the aid of the fewest and simplest general formulæ. Why things happen as they do, it is now said, is no proper question for science; its sole business is to enable us to calculate how they will happen. With the general epistemological questions raised by this doctrine we must deal later in our third and fourth books. At present we are concerned only with its bearing on the notion of Causal Relation.[[107]]
The important point for our immediate purpose is that the reduction of all events to continuous processes really does away with Causation altogether, as is recognised by those adherents of the theory who openly propose to expel the word “cause” from the language of science.[[108]] For in a continuous process it is purely arbitrary where we shall mentally draw the dividing line which is to mark the boundary between the “earlier” and the “later” stage. What the descriptive formula, with the aid of which we trace the course of the process by giving a series of successive values to our time-variable presents, is not the “cause” of the process but the “law” of it. Instead of looking upon the later stages of the process as determined by the earlier, we are now looking upon the process as a whole as the expression in detail of a single principle. We have, in fact, abandoned the category of cause and effect for that of Ground and Consequence. We are seeking the ground of the whole process not in a set of temporally preceding events, but in its own pervading principle.
From this point of view the one-sided dependence of effect on cause, characteristic of the causal relation, disappears. Whether we shall infer the later stages of the process from the earlier, or the earlier from the later, depends simply upon our choice of positive or negative values for our time-variable. For “descriptive” science, what we suggested at first as a paradoxical possibility is the actual fact. The past is determined by the future in precisely the same sense in which the future is determined by the past, namely, that as both are stages of the same continuous process, if once you know the principle of the process you can start equally well with either and reason to the other.[[109]] Thus, within the limits of experimental science itself, the conception of causal relation has given way to the conception of events as logically connected into a system in virtue of their underlying ground or principle. For practical purposes experimental science has, in its application of this conception, to be guided by two postulates, neither of which can be metaphysically justified. It has to assume (a) that the course of events is composed of a plurality of more or less independent continuous processes, each of which has its own ground within itself, at least to such an extent as to be capable of being treated for our purpose as independent of others; (b) that the underlying ground or grounds of all events can be adequately expressed in terms of mathematical symbolism.
As to the first of these points, our discussion of the unity of Reality convinced us that there must in the end be a single ground of all existence, and therefore the complete reason of any partial process cannot be entirely within itself. The independence of the various processes must be relative, and even the belief that it is sufficient to enable us to treat them for our own special purposes as self-contained and independent, must be a postulate prompted by our practical needs, and justified in the end by its success. The second point will engage our attention more fully in subsequent chapters. At present one remark upon it must suffice. The calculability of the laws of continuous processes depends upon our ability to reduce them to numerical and quantitative forms. Wherever we have the appearance, at any stage in a process, of a new quality, we have in fact an apparent breach of continuity, and it ceases to be in our power to exhibit the new stage of the process as a mere transformation of what was already expressed in former stages. Hence the success of natural science in reducing all sequences of events to continuous processes depends upon the assumption that we can establish equations between qualitatively different magnitudes.