Natural Philosophy

by Wilhelm Ostwald

Let us picture to ourselves some process of measurement—for example, the determination of the length of a strip of paper. We place a rule divided into millimeters (or some other unit) on the strip, and then we determine the unit-mark at which the strip ends. It turns out that the strip does not end exactly at a unit-mark, but between two unit-marks. And even if the rule is provided with divisions ten or a hundred times finer, the case remains the same. In most cases a microscopic examination will show that the end of the strip does not coincide with a division. All that can be said, therefore, is that the length must lie between n and n + 1 units, and even if a definite number is given, the scientifically trained person will supplement this number by the sign ± f, in which f denotes the possible errors, that is, the limit within which the given number may be false.

We see at once how the characteristic concept of threshold, which has led to the conception of the continuous, immediately asserts itself when in connection with discontinuous numbers. The adaptation of the threshold to numbers can be carried as far as it is possible to reduce the threshold, but the latter can never be made to disappear entirely.

The significance of measurement therefore lies in the fact that it applies the operation of counting with all its advantages (see [p. 85]) to continuous things, which as such do not at first lend themselves to enumeration. By the application of the unit measure a discontinuity is at first artificially established through dividing the thing into pieces, each piece equal to the unit, or imagining it to be so divided. Then we count the pieces. When a quantity of liquid is measured with a liter this general process is carried out physically. In all other less direct methods of measurement the physical process is substituted by an easier process equally good. Thus, in the example of the strip of paper we need not cut it up into pieces a millimeter in length. The divided rule is available for comparing the length of any number of millimeters that happen to come under consideration, and we need only read off from the figures on the rule the quantity of millimeters equal to the length of the strip, in order to infer that the strip can be cut up into an equal number of pieces each a millimeter in length.

After it has been made possible to count continuous things in this way, the numeration of them can then be subjected to all the mathematical operations first developed only for discrete, directly countable things. When we reflect that our knowledge of things has given them to us preponderatingly as continuous, we at once see what an important step forward has been made through the invention of measurement in the intellectual domination of our experience.

37. The Function.

The concept of continuity makes possible the development of another concept of greater universality, which can be characterized as an extension of the concept of causation ([p. 31]). The latter is an expression of the experience, if A is, B is also, in which A is understood to be a definite thing at first conceived of as immutable. Now it may happen that A is not immutable, but represents a concept with continuously changing characteristics. Then, as a rule, B will also be of that nature, so that every special value or state of B corresponds to every special value or state of A.

Here, in place of the reciprocal relation of two definite things, we have the reciprocal relation of two more or less extended groups of similar things. If these things are continuous, as is assumed here (and which is extremely often the case), both groups or series, even though they are finite, contain an endless quantity of individual cases. Such a relation between two variable things is called a function. Although this concept is used chiefly for the reciprocal relation of continuous things, there is nothing to hinder its application to discrete things, and accordingly we distinguish between continuous and discontinuous functions.

The intellectual progress involved in the conception of the reciprocal relation of entire series or groups to one another, as distinguished from the conception of the relations between individual things, is of the utmost importance and in the most expressive manner characterizes the difference between modern scientific thought and ancient thought. Ancient geometry, for example, knew only the cases of the acute, right, and obtuse angled triangle, and treated them separately, while the modern geometrician represents the side of the triangle as starting from the angle zero and traversing the entire field of possible angles. Accordingly, unlike his colleague of old, he does not ask for the particular principles bearing upon these particular cases, but he asks in what continuous relation do the sides and angles stand to one another, and he lets the particular cases develop from out of one another. In this way he attains a much profounder and more effectual insight into the whole of the existing relations.

It is in mathematics in especial that the introduction of the concept of continuity and of the function concept arising from it has exercised an extraordinarily deep influence. The so-called Higher Analysis, or Infinitesimal Analysis, was the first result of this radical advance, and the Theory of Functions, in the most general sense, was the later result. This progress rests on the fact that the magnitudes appearing in the mathematical formulas were no longer regarded as certain definite values (or values to be arbitrarily determined), but as variable, that is, values which may range through all possible quantities. If we represent the relation between two things by the formula B = f(A), expressed in spoken language by B is a function of A, then in the old conception A and B are each individual things, while in the modern conception A and B represent an inexhaustible series of possibilities embracing every conceivable individual case that may be co-ordinated with a corresponding case.

Herein lies the essential advantage of the concept of continuity. It is true that it also introduces into calculation the above-mentioned contradictions which crop up in the ever-recurring discussions concerning the infinitely great and the infinitely small. The system introduced by Leibnitz of calculating with differentials, that is, with infinitely small quantities, which in most relations, however, still preserve the character of finite quantities from which they are considered to have been derived, has proved to be as fruitful of practical results as it is difficult of intellectual mastery. We can best conceive of these differentials as the expression of the law of the threshold, which law gave rise to, or made possible, the relation between the continuous and the discrete.