Natural Philosophy

by Wilhelm Ostwald

38. The Application of the Functional Relation.

I have already shown ([p. 34]) how the first formulation of a causal relation which experience yields can be purified and elaborated by the multiplication of the experience. The method described was based upon the fact that the necessary and adequate factors of the result were obtained by eliminating successively from the "cause" the various factors of which its concept was or could be compounded, and by concluding from the result, that is, the presence or absence of the "effect," as to the necessity or superfluity of each factor.

Obviously the application of this process presupposes the possibility of eliminating each factor in turn. Very often it is not possible, and then in place of the inadequate method of the individual case the method of the continuous functional relation steps in with its infinitely greater effectiveness. If in most cases we cannot eliminate the factors one by one, there are very few instances in which it is not possible to change them, or to observe the result in the automatically changed values of the factors. But then we have the principle that for the causal relation all such factors are essential the change of which involves a change of the result.

It is clear that this signifies a generalization of the former and more limited method. For the elimination of the factor means that its value is reduced to zero. But now it is no longer necessary to go to this extreme limit; it suffices merely to influence in some way the factor to be investigated.

It is true that here the difference in the result cannot be expressed with a "yes" or a "no," as before. It can only be said that it has changed partly, more or less. From this it can be seen that the application of this process requires more refined methods of observation, especially for measuring, that is, for determining values or magnitudes. On the other hand, we must recognize how much deeper we can penetrate into the knowledge of things by the application of the measuring process. Each advance in precision of measurement signifies the discovery of a new stratum of scientific truth previously inaccessible.

39. The Law of Continuity.

From the fact that natural phenomena in general proceed continuously we can deduce a number of important and generally applicable conclusions which are constantly used for the development of science.

When a relation of two continuously varying values of the form A = f(B) is conjectured, we convince ourselves of its truth by observing for different values of A the corresponding values of B, or reversely. If we find that changes in the one correspond to changes in the other, the existence of such a relation is proved, at first only for the observed values, though we never hesitate to conclude that for the values of A lying between the observed values, but themselves not yet observed, the corresponding values of B will also lie between the observed values. For example, if the temperature at a given place has been observed at intervals of two hours, we assume without hesitancy that in the hours between when no observations were made, the values lie between the observed values. If we indicate the time in the usual manner by horizontal lines and the temperature for the general periods of time by longitudinal lines, the law of continuity asserts that all these temperature points lie in a steady line, so that when a number of points lying sufficiently near one another is known, the points between can be derived from the steady line which may be drawn through the known points. This very commonly applied process will yield the more accurate results the nearer the known points are to one another, and the simpler the line.

The application of the law of continuity or steadiness, therefore, means no less than that it is possible, from a finite, frequently not even a very large, number of individual results, to obtain the means of predicting the result for an infinitely large number of unexamined cases. The instrument derived from this law, therefore, is an eminently scientific one.

The value of this instrument is still greater if it succeeds in expressing the relation A = f(B) in strict mathematical form. First, the result of the determination of a number of individual values of that function is represented as a table of co-ordinated values. By the graphic process above described, or by its equivalent, the mathematical process of interpolation, this table is so extended that it also supplies all the intermediate values. But this is still a case of a mechanical co-ordination of the corresponding values. Often we succeed, especially in the relation of simple or pure concepts, in finding a general mathematical rule by which the magnitude A can be derived from the magnitude B, and reversely. This is the only instance in which we speak of a natural law in the quantitative sense.