In this connection it is most important to understand the difference between a physical and a mathematical definition. Take a number like n in mathematics. We can define it as the ratio of the length of a circumference to its diameter in Euclidean geometry. Without performing physical measurements, we can deduce from this definition, by purely mathematical means, the precise value of
to any order of approximation we please. Thus the definition does not lead to ambiguity, hence is a valid one.
On the other hand, try to give a physical definition by some similar method—say, the definition of the “gram” or of the “dyne.” The purely logical type of definition breaks down, and we are compelled to resort to physical determinations. Accordingly, we define the gram as the mass of a cubic centimetre of distilled water under specified conditions of temperature and pressure.
And why was the purely logical definition possible in mathematics and a failure in physics? Merely because in mathematics—in the case of
, for example—the laws of Euclidean geometry were accepted from the start, whereas in physics preliminary information of this sort is denied us. And it is the same with all the magnitudes and constants of physics. It is impossible to predetermine by logical means the value of the invariant velocity, or of the gas constant, or of the size of the earth or of an electron. In every case, we must have recourse to physical measurements of one sort or another, for the constants of physics present no character of rational necessity.
Much of the criticism directed against Einstein’s use of rods, clocks and light propagation arises from a failure to understand this fundamental point. Purely logical, non-empirical definitions in physics can only lead to ambiguity. As for the theory of relativity, it is one of physics; it does not claim or aspire to be anything else; and a theory of physics which would avoid physical measurements is about as reasonable a conception as a forest without trees.
CHAPTER XX
THE IRREVERSIBILITY OF TIME
WE saw, when discussing the existence of a finite invariant velocity in nature, that the presence of this velocity was going to work havoc with our belief in absolute simultaneity. Thus, if two events take place in different places, we can no longer attribute any universal meaning to the opinion that these two events have taken place at one and the same instant of time or at two different instants. In other words, there exists no absolute clock giving us the correct time at all points of the universe. According to the motion of the observer, two events which are simultaneous for one may follow in sequence for another.