Here, of course, it might be argued that we are unwittingly introducing an assumption when we state that beings who neglected to conform to the same rules of reasoning would fail to agree. Of course, if we adopt so hypercritical an attitude, there is nothing left to argue about. The instant we start to talk we are assuming from past experience that sounds will be emitted and that our words will convey at least some meaning to our neighbour. It is the same with the rules of logic. Regardless of what the ontological status of these rules may be, we may assume that in view of past experience it is convenient for men to abide by them if they wish to construct a common science.
Let us consider now what scientists mean by the uniformity of nature. The uniformity of nature may be interpreted in various ways. It may mean that nature is regulated by universal laws. In this form, it is obviously not a presupposition adopted uncritically. The primitive man does not feel the necessity for it; he prefers to believe in miracles. If the modern physicist accepts this uniformity it is because it has been confirmed a posteriori in so large a number of cases that it seems simpler to accept it than to reject it, at least until further notice. Furthermore, it is an assumption which is perfectly well recognised as of limited validity, and scientists are the first to suspect that after a more microscopic survey of nature it may prove untenable. Nature may not be rational and our logic may be unable to cope with it. If this be the case, statistical knowledge may be the only type possible, though even statistical knowledge may fail us.
We can also understand the uniformity of nature in a different sense. We may wish to imply that an experiment attempted on a Monday would yield the same result if repeated on a Tuesday. In other words, the passage of time, though affecting us in a number of ways (aging, etc.), can be disregarded in certain important cases. Suppose, for the sake of argument, that we were to dispute the legitimacy of this assumption; what would be the result? Obviously science would become impossible. A geometrical proof holding in Euclid’s time would no longer hold to-day; knowledge would be intransmissible. There would be no object in performing an experiment to-day, since to-morrow our conclusions would have lost all value. In fact, it would be futile to attempt any experiment at all, since every experiment is spread over a certain time. In a world of this sort, prevision would be quite impossible; a rule of action would be out of the question. Inasmuch as one of the most important functions of science is to serve us with a rule of action, we can realise that science would become unthinkable. This brings out an important difference between science and history, for example. History tells us of events past and gone which, for all we care, may never reoccur; but in science a phenomenon which could never occur again would be of little interest. In short, we must agree that the very existence of a common science proves that our assumption of the uniformity of nature, whether it be justified or not sub specie œternitatis, is justified at least in that limited field of experience which is ours. On no account, then, must our belief be confused with a presupposition adopted uncritically.
As a matter of fact, even were we assured that the uniformity of nature was a mistaken belief, we should continue to be governed by it just the same. We know perfectly well that the uniformity of our stream of terrestrial life does not hold; we know that though we have risen every day, yet at any instant we may die. But unless we wish to exist in a state of complete stagnation, we must act as though our life would continue, and so we plan for the future. Under the circumstances, it appears futile to seek any rational justification for the uniformity of nature, basing our arguments on probability considerations. The fact that things have existed until to-day does not prove they will exist to-morrow; but it suggests that we act as though they would.
Among the principles, special mention must be made of the “principle of sufficient reason.” In physical science, the application of this principle is subject to all manner of difficulties. For instance, when we assert that there is no reason for a body to move here rather than there, we are assuming that we possess at least a general knowledge of all the possible influences which happen to be present.
Obviously, our assumption is apt to lead us to erroneous conclusions. We mentioned a case in point when discussing the problem of space. Assuming space to be perfectly empty, the principle of sufficient reason suggested that it should be homogeneous and isotropic; but as we were by no means certain that physical space was truly empty, the application of the principle was of no great value.
Difficulties of this sort are by no means restricted to problems of physics. Even in pure mathematics it is not always easy to decide in what way the principle should be applied. The mathematical problems in which we are compelled to appeal to sufficient reason are those dealing with probabilities; and the following elementary illustration, discussed by Bertrand, will afford us a sample of the type of difficulties which we may encounter. Suppose, for instance, we are asked: “What is the probability that on tracing the straight line at random, the line-segment contained within a given circle should be greater than the side of the equilateral triangle inscribed in the said circle?”
Now, if we conceive of all possible straight lines intersecting the circle, sufficient reason tells us that any one of these lines is as likely to be traced as any other. Hence, the probability we are seeking will be given by the ratio of the number of different lines that satisfy our stipulated condition, to the total number of different lines that can be drawn to intersect the circle. Clearly, this last number will be given by considering all the lines passing through a given point of the circumference and then conceiving this point as occupying in succession every position on the said circumference. Under these conditions the probability turns out to be ⅓. But we might, with equal justification, have proceeded in another way. We might have considered all the straight lines parallel to a given direction and then conceived of this direction as suffering in succession every possible orientation. In this case, the probability would work out as ½. This example warns us that a large measure of arbitrariness lies concealed even in the most elementary problems where sufficient reason is appealed to.
As for the regulative norms which guide scientific discovery, we have seen that they reduce to a search for unity in diversity; to a desire to generalise, to detect uniformities or laws, and thus to economise effort, as Mach has aptly said. These same regulative norms are just as apparent in mathematics as in physics and have been heeded by all scientists from Democritus to the moderns.
We might also mention the principle of continuity and the principle of causality as understood by modern scientists. The principle of continuity implies the eradication of all action at a distance such as is exemplified in Newton’s law of gravitation. It is probable that a belief in the principle of continuity arises from the fact that in daily life all action seems to be transmitted by contact. Again, an object moves from “here” to “there” through a continuous series of intermediary positions, or at least when such is not the case we cease to recognise it as the same object. Newton himself felt the necessity of adhering to the principle of continuity, but in his day the development of theoretical science was not sufficiently advanced to permit a realisation of his hopes. For the same reason, physicists during the eighteenth and nineteenth centuries paid little heed to the principle. Particles were assumed to attract one another according to certain laws, but how the attraction was transmitted was never discussed.