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An Introduction To Non-aristotelian Systems And General Semantics.

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written If we were to follow the operation Q by the differential operation
the result would be*:
But as / is arbitrary, it ,may be omitted from the equations and the result written in the operator form as
The symbol, denned by this equation, should be read as 'the operation of
multiplying by. Similarly,
In translating the ordinary equations into the language of operators nothing new is introduced. This translation involves only a change of 'mental' focus. Instead of concentrating our attention on the numerical values we concentrate on the operations of combining them. Since the great problems of the quantum mechanics consist in finding new methods of computation, or of combining numerical values, such a change of attitude may prove to be structurally useful.
It should be noticed here that once matrices are considered, and treated, as quantities, or unique and specific relations, by similar reasoning they can be treated as operators. This problem is of fundamental structural and semantic importance because in the quantum theory we deal with matrices which have infinite numbers of terms and since this complexity presents great technical difficulties it is of enormous advantage to be able to pass to some more developed methods of calculation.
At the preliminary stage in the operator calculus we have assumed that multiplication was commutative, that QP=PQ; but in the further and more general development of the theory, this does not hold.
In general we must assume in the operator calculus that multiplication is not commutative, that For instance, ifand, the
two operations are certainly not commutative. Naturally, the validity of 2 X2 =4 is not doubted, but generalized non-commutative multiplication has a definite asymmetrical and so structural geometrical interpretation, to be found in the vector calculus. When we associate with each numerical quantity its own operation of multiplication we thus obtain a more general calculus. Operators may be regarded as compound, or built up from the elementary arithmetical operations of addition and multiplication. They represent, so to say, functions of these operations.7
The operator Dx, also written d/dx, is called a differential operator. If applied to a product (w») the results are given by the formula