712 X. ON THE STRUCTURE OF 'MATTER'
but on the semantic operations of combining them. The calculi used in the newer quantum mechanics are peculiar, because, while they retain the numerical data, and as far as possible, the classical equations, they alter the operations by which these quantities are combined, or the interpretation of the equations. As an illustration of such a procedure we can take two different formulae for the addition of velocities) one from the classical mechanics, where
(1) and the other, the formula for velocity as given by the Einstein theory; namely,
In these formulae represents the velocity of body 1 relative to body 2 ,. In formula (1) the sign ' +' symbolizes the ordinary arithmetical operation of addition. As we already know, this formula has proven too simple to represent accurately the experimental data, and Einstein has replaced it by the more elaborate formula (2).
The above statement is the usual way of speaking about the modification in formulation which has taken place in physics since Einstein. But we could equally well say that the formula has not been altered except that the '+' has no longer the old meaning and does not now represent the arithmetical operation of addition. Both points of view lead ultimately to one value, , and the computations are similar in both cases.
We should notice especially the great freedom with which we can treat mathematical entities. Our voluntary selection of the point of view becomes important. A similar freedom of selection of interpretation appears to a still larger extent in all verbal problems, a fact of considerable structural and semantic importance in any theory of sanity, as we have already seen.
Further illustration of this freedom can be seen in the way in which the ordinary notion of multiplication is re-interpreted in the operator calculus. Let us denote by q and /, two numerical quantities, and qf as their product. But we could view this problem differently. We could say that qf results from a semantic operation q performed on /, or a semantic operator (q X) acting upon / which transformed / into qf. We could denote the operation of multiplying by q or the operator (q X) by a single symbol Q. Quite obviously the operator Q is not the number q; in other words, the semantic operation of multiplying, say by two, is not the number 2.
The operation of multiplying integer 1 by integer 2 gives the result 2. Similarly, the operation of multiplying 1 by q, or, in our new language, the application of the operator Q to the integer 1, gives q. In symbols, Ql =q. If we take any arbitrary function /, the result of the operation of Q upon / is