36. Measurement.
Measuring is in a certain way the opposite of counting. While, in counting, the things are regarded in advance as individual, and the group, therefore, is a body compounded of discontinuous elements, measuring, on the other hand, consists in co-ordinating numbers with continuous things, that is, in applying to continuous things a concept formed upon the hypothesis of discontinuity.
It lies in the nature of such a problem that the difficulty of adaptation must crop out somewhere in the course of its attempted solution. This is actually shown by the fact that measurement proves to be an unconcluded and inconcludable operation. If, in spite of this, measurement may and must justly be denoted as one of the most important advances in human thought, it follows that those fundamental difficulties can practically be rendered harmless.
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.