Thus we see that it seems necessary to draw a marked distinction between two types of calculability. Calculation based on insight into individual character and purpose is so far from being inconsistent with purposiveness and intelligence, that the more coherent and systematic the purposes by which a life is controlled, the more confident does such calculation become. Calculation without such special knowledge, and based upon mere general propositions, on the other hand, cannot be regularly successful where one has to deal with the behaviour of individual purposive beings.[[129]]
Now, the difficulty as to our interpretation of the physical order as the presentation to our sense of a system of intelligent purposive beings, is that the successes of physical science seem at first sight to show that just this “mechanical” calculation of the course of events from observed sequence, without insight into underlying individual purpose, is possible when we are dealing with physical nature. For, on the one hand, we ourselves admitted that if physical nature is permeated by individual purposes, we do not know what those purposes in detail are; and, on the other, it is undeniable that physical science, which systematically disregards their presence, has been signally successful in the past, and may be expected to be even more successful in the future, in detecting uniformities in physical nature, and so submitting it to exact calculation. Hence it might be thought that the actual success of the empirical sciences cannot be reconciled with the principles of our metaphysical interpretation of the course of nature.
We must, however, draw a very important distinction. There is one method by which uniformities of a certain kind can be detected in the behaviour of purposive intelligent beings, without insight into the nature of their individual purposes—the method of statistical averages. Thus, though it would be quite impossible to say with certainty of any individual man that he will shoot himself or will get married, except on the strength of insight into his individual character and interests, we find by experience that it is possible to say, within a certain narrow range of error, what percentage of Englishmen will shoot themselves or will get married in the year. The percentage is, of course, rarely or never precisely realised in any one year, but the longer the period of years we take for examination, the more exactly do the deviations from the average in individual years compensate one another. The explanation is, of course, that on the whole the incentives to marriage or suicide, in a reasonably stable state of society, remain constant from year to year, so that by taking an average of several years we can eliminate results which are due to individual peculiarities of temperament and situation, and obtain something like a measure of the degree in which the general conditions of social existence impose a certain common trend or character on the interests and purposes of individuals.
Two things are at once noticeable in connection with all uniformities obtained by the method of averages. One is that the result formulated in the statistical law is always one to which the actual course of events may reasonably be expected to conform within certain limits of deviation, never one to which we have a right to expect absolute conformity. Not only is the actual number of marriages, e.g., in any one year, usually slightly above or below the average percentage computed, e.g., for a ten years’ period, but as we compare one longer period with others, the average percentage for the longer period itself fluctuates. It is only in the “long run,” that is, in the impossible case of the actual completion of an interminable series, that the computed average would be exactly realised. As every one who has to deal with averages in any form knows, precise realisation of the computed average within a finite series of cases would at once awaken suspicions of an error somewhere in our calculations. Thus the uniformities of this kind are never absolutely rigid; they are ideal limits to which the actual course of events is found to approximate within certain limits of divergence.
The second point is that the existence of such a uniformity never affords logical ground for confident affirmation as to the actual event in a particular concrete case. To revert to our illustration, just as we have no right to infer from the approximately constant percentage of marriages per year in a given society, that this precise percentage will be realised in any one special year, so we have still less right to infer that a particular member of that society will or will not marry. Nothing but insight into the character, situation, and interests of this special member of society can give me the right to judge with confidence how he will actually behave. Similarly, it is possible to say within certain limits of error how many persons over sixty years of age may be expected to die in the next twelve months, but it would be the height of logical presumption to infer that a particular man will die during the year, except on the strength of special information about his pursuits, habits, and general state of health.[[130]] Thus our general conclusion must be, that calculation and the establishment of uniformities is possible, without insight into individual purpose, but that the uniformities thus obtained are always variable and approximate, and afford no safe ground for inference as to special concrete cases.
§ 3. (2) Uniformity in Physical Nature. The existence of ascertainable uniformities in physical nature, then, will not conflict with our general interpretation of the physical order, provided that these uniformities are of the type just illustrated by reference to ordinary social statistics. On the other hand, the exact and rigid conformity of the actual course of concrete events with such uniform general “laws,” would certainly be inconsistent with the presence of teleological adaptation to ends. A reign of rigid routine conformity to general law cannot co-exist with individual purposive life. Now, it is commonly assumed, and we shall shortly see that the assumption is both necessary and justified as a practical methodological postulate, that the “reign of law” in physical nature is absolute. But are there any grounds for recognising this assumption as more than a possibly unrealised postulate made for human practical purposes? I think it is easy to show that there are none whatever, and that the conception of a nature devoid of purpose and sentience, and swayed absolutely by mechanical “laws,” is simply a metaphysical nightmare of our own invention.
To begin with, it is clear that the undeviating conformity of the actual course of any concrete process to scientific “law” cannot be verified as an empirical fact by observation or experiment. For in no observation or experiment can we ever deal with the whole of any concrete actual event or process. We have always, for the purposes of our observation, to select certain of the general aspects of the process, to which we attend as the “relevant factors” or “conditions” of the result, while we disregard other aspects as “immaterial” or “accidental” circumstances. And this artificial abstraction, as we saw in discussing Causality, though indispensable for our practical purposes, is logically indefensible. Again, within the aspects selected for attention, all that experiment can establish is that the deviation from uniform law, if there is any deviation, is not sufficiently great to affect our measurements and calculations. But how far our standards of measurement are from rigid precision may be readily learned from the chapter on physical standards in any good work on the logic of the inductive sciences.[[131]] Our failure to detect deviation from law is absolutely worthless as evidence that no deviation has taken place.
Thus, if the absolute uniformity of natural processes is more than a practical postulate, it must be an axiom, that is, it must be implied in the very notion of those processes as elements in a systematic whole. But it should at once be clear that we have no more ground for asserting such uniformity as an axiom, than we had for treating the causal postulate as axiomatic. It is by no means implied in the concept of a systematic whole that its parts shall be connected by uniform law. For the unity of the system may be teleological, that is, the parts may be connected by the fact that they work together to realise the same end, to execute the same function. In that case the behaviour of any one part will depend on the demands laid upon it by the plan which the working of the system fulfils. As these demands vary from time to time, the behaviour of the part under consideration will then vary correspondingly, though to all appearance its surroundings may, for a spectator who fails to grasp the end or purpose realised by the system, be identical.[[132]] This is actually the case with those systematic wholes in which human insight can directly detect unity of purpose or aim. A man with definite purposes before him does not react in a uniformly identical way upon situations which, apart from their relation to his purpose, would be pronounced identical. He learns, for instance, from previous failure in the same circumstances, and so acquires the power to react on them in a way better adapted to the obtaining of his end. Or his progressive execution of his purpose, where there has been no failure, may require different conduct on the two occasions. To speak with strict logical accuracy, the situations, relatively to his special purpose, are never identical, though it may be no difference could be detected in them apart from that relation to this peculiar purpose.[purpose.] Relatively to the system of intelligent purposes which realises itself through the circumstances, every situation is, properly speaking, unique.
Now, if we consider the methods by which the uniformities called “laws” of nature are actually formulated, we may see ground to conclude that they may one and all be uniformities of the approximate non-exact type. In many cases, if not all, these uniformities have manifestly been obtained by statistical methods. Thus, for example, when it is said that all the atoms of a given chemical element are absolutely alike, e.g., that every atom of oxygen has the atomic weight 16, there is absolutely no valid ground for regarding this uniformity as actually realised without deviation in individual cases. If the atom should prove, as it may, to be no more than a convenient device of our own, useful for computing the behaviour of sensible masses but with no real existence of its own, it is of course evident that there can be no question of the real conformity of individual cases to the law. But even if there really are indivisible bodies answering to our conception of atoms, still we have to remember that we have no means of dealing with the individual atom directly. We infer its properties indirectly from the behaviour of the sensible masses with which we can deal more directly. Hence, at most, the statement that the atom of oxygen has a certain weight means no more than that we can for our practical purposes disregard any possible individual divergences from this value. The oxygen atoms, if they really exist, might actually fluctuate in individual atomic weight about an average; yet, so long as we cannot deal with them individually but only in bulk, these fluctuations, if only sufficiently small, would produce no appreciable effect on our results, and would therefore properly be treated in our science as non-existent. Conceivably, then, such chemical uniformities may afford no safer ground for precise statements about the weight of the individual atom, than anthropological statistics do for precise statement about the actual height, weight, or expectation of life of an individual man. And we can readily see that a non-human observer with senses incapable of perceiving the individual differences between one man and another, might be led from the apparent uniformity of behaviour exhibited by large collections of human beings to the same sort of conclusions which we are tempted to make about atoms.[[133]]
Similarly with other cases of apparently rigid uniformity. As any one who has worked in a laboratory knows,[[134]] such results are in actual practice obtained by taking the mean of a long series of particular results and treating the minor divergences from this mean as non-existent because they are negligible for all practical purposes. In other words, the apparently rigid conformity of natural processes to uniform law is an inevitable consequence of the fact that we are debarred by various limitations of a subjective kind from following the course of any process in its individual detail, and have therefore to make all our inferences from the observation and comparison of series of processes sufficiently extended for individual differences to neutralise each other. But in all this there is absolutely no warrant for the conclusion that the course of any one individual process is absolutely uniform with that of any other. There is room within the uniformity got by these methods of comparison for an infinite variety of individual detail, of which our scientific constructions take no account, either because our means of observation are insufficient to detect it, or because, when detected, it is of no significance for the original object of our science—practical success in interference with the course of events.