Numbers and measurement

The concept of measurement lies midway between experiment and theory. To measure is to assign a number to a property of an object (length, volume…), to a sensation (temperature…), or even to something invisible (an electric current…). Defining it, however, requires a coherent scientific theory. The first step is to choose a scale—that is, an origin and an increment (a unit). In our current understanding, therefore, a measurement is a number accompanied by a unit. It would be a mistake to confuse the two concepts.

Quantities and measurements --------------------

The ancients drew a clear distinction between the concept of "number"—by which they meant integers—and that of "quantity." They often used the latter term for a length, but also for an area or a volume. Building on ideas developed by Eudoxus of Cnidus, Euclid considered two quantities of the same kind to be commensurable if repeating the first some number n of times made it coincide exactly with the second, itself repeated some number p of times. Today, taking the first as our standard, we would say that the second measures n/p times the first. The concept underlying this idea is really that of a fraction, but the ancients did not regard a fraction as a number at all, seeing it instead as a comparison between numbers. Thus, two quantities are commensurable if one can be measured using the other as a standard. This is not automatic: a square's diagonal is incommensurable with its side, a fact that had plunged the Pythagoreans into deep perplexity…