The uniformity of nature, as the scientist understands it, is much more than th:s. It could be, called more appropriately the continuity of nature. Perhaps it is best to consider it in the form which it assumes in the Newtonian physics. In the Newtonian physics, the world is completely described when the density of the matter occupying each point of space at some instant is known, together with the magnitude and direction of its velocity. The investigations of physics consist then in determining the actual form of the relation between quantities repre senting time, space, local density, direction and magnitude of velocity. In the attempts to dis cover the function which the time forms of the remaining seven variables, one assumption is always made— that this function is in general (i.e., in the language of the mathematician ex cept at a set of points of zero measure) what the mathematicians term analytic. One conse quence of this is that the function is continuous — i.e., by making sufficiently small changes in the seven variables, we may make the difference of time that results smaller than any assigned quantity, and keep it smaller. Furthermore, if we take a large number of experimental deter minations of the time and the other seven vari ables, it is possible to construct a unique func tion of the seven variables which will represent the time at each of these points and which, by simply increasing the number of experiments, may be made to differ from the function repre senting the actual course of phenomena by less than any assigned quantity over any desired period of time. Two things follow: first, sufficiently slight errors in the observations only mean slight errors in the law cover ing the observations; secondly, by increas ing the number of observations, it is pos sible to render the maximum error in the law formulated to cover them less than any given value. These facts assure us that by taking a sufficient number of observations, and by exer cising a sufficient amount of care in each obser vation, we may approximate as near to the truth as we desire. That the amount of labor in obtaining a reasonable approximation is not be yond all human powers is an article of faith which may be said to constitute a part of what we mean by the law of the uniformity of nature. Other important elements have to do with the spatial distribution of phenomena. Not only the very small, but also the very remote, is inac cessible to observation and measurement. The existence of a scientific physics depends on our ability to neglect the phenomena at sufficiently great astronomical distances. Similar proposi tions assuring us of the negligibility of that which is sufficiently difficult to observe are basal for other aspects of physical theory.
The law of the uniformity of nature in physics then involves something like the follow ing aggregate of statements: (1) There is a single equation subsisting between the time, the density of matter at every point of space at the time, and the direction and magnitude of veloc ity at thatpoint. (2) This equation is such that each of the unknowns involved is in gen eral an analytic function of all the rest—that is, among other things,. that sufficiently small i changes in the other unknowns produce only slight changes • therein. (3) The error as to the occurrences near us made by con sidering only those physical phenomena within a sphere of finite radius with ourselves at the centre ultimately decreases very fast as the radius of the sphere increases.
If these statements are taken together with rough estimates of the order of magnitude of the change that the changes of some of the physical phenomena mentioned in 2 and 3 entail in others, we have the barest outline of the law of the uniformity of nature as it is found in physics. This law is very far from
being a tautology — it is not even obvious. Furthermore, it is specifically a law of physics, and has only been established since the time of Newton by inductive physical 'researches. Other disciplines, such as psychology, have related but different laws of uniformity. They all involve statements of the continuity of certain concrete phenomena. It appears, then, that induction de mands antecedent universal propositions that are not identically true — ex puns particularibus nil concluditur is not confined to deduction. These synthetic propositions, a priori at least in part, go back and back until in the last analysis they are due simply to a general consonance between the human mind and the facts of nature. This consonance, which consists largely in a prefer ence for continuity both on the part of the mind and of nature, is continually rendered more perfect by the attrition of our imaginings in the places where they disagree with our observa tions. The history of science consists in a gradual remodeling of each theory in the points where it is wrong, in a mathematical treatment of the errors of the last mathematical treat ment. It will be seen that in the theory here developed the distinction between induction and deduction is not absolute, but is rather one of degree and attitude. The stages of an induc tive research are: (1) the imagination of a theory to fit the facts; (2) the deduction of the consequences of the theory; (3) the verifi cation of these consequences and the observa tion of their errors; (4) the imagination of a theory to account for the errors of the original theory or the formulation of a new theory avoiding these errors. The process runs through a regular never-ending cycle. Stages (1) and (2) are identically those of the mathematician in his purely deductive reasoning. Stages (3) and (4) may be paralleled in a mathematical re search where the object is the formation of an algorithm which will subserve an especial end. The only difference is that the verification which the mathematician makes is complete, that of the physicist incomplete.
The importance given to continuity in this article may he expressed by saying that the chief inductive method of the scientist is what Mill calls the method of concomitant variations. Mill's canon for this method is: °Whatever phenomenon varies in any manner wherever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.° It will be noted that Mill gives a causal interpretation to the method. It has always been the custom of the philosopher, and almost never the custom of the scientist, to interpret the laws of nature under the aspect of cause.
A law of nature is simply a more or less precise formula to which occurrences conform. Some times, and only sometimes, the correlated phenomena will have a temporal order, and we may talk of antecedents and consequents. In such a case the antecedents may be called causes and the consequents effects. This implies no obscure effective force emanating from the cause and proceeding to the effect — Hume de molished that notion long ago. A causal in terpretation of the universe, then, consists merely in selecting one especial type of induc tive correlation and elevating it to the type of all induction whatever.