Now we can get rid of all repetitions of fractions of the same value by simply striking them out whenever they appear after their first occurrence. In the few initial terms written down above, which is enclosed above in square brackets is the only fraction not in its lowest terms. It has occurred before as . Thus this must be struck out. But the series is still left with the same properties, namely, (a) there is a first term, (b) each term has next-door neighbours, (c) the series goes on without end.
It can be proved that it is not possible to
arrange the whole series of real numbers in this way. This curious fact was discovered by Georg Cantor, a German mathematician still living; it is of the utmost importance in the philosophy of mathematical ideas. We are here in fact touching on the fringe of the great problems of the meaning of continuity and of infinity.
Another extension of number comes from
the introduction of the idea of what has been variously named an operation or a step, names which are respectively appropriate from slightly different points of view. We will start with a particular case. Consider
the statement . We add to and obtain . Think of the operation of adding : let this be denoted by . Again . Think of the operation of subtracting : let this be denoted by . Thus instead of considering the real numbers in themselves, we consider the operations of adding or subtracting them: instead of , we consider and , namely the operations of adding and of subtracting . Then we can add these operations, of course in a different sense of addition to that in which we add numbers. The sum of two operations is the single operation which has the same effect as the two operations applied successively. In what order are the two operations to be applied? The answer is that it is indifferent, since for example so that the addition of the steps and is commutative.
Mathematicians have a habit, which is puzzling to those engaged in tracing out meanings, but is very convenient in practice, of using the same symbol in different though allied senses. The one essential requisite for a symbol in their eyes is that, whatever its possible varieties of meaning, the formal laws for its use shall always be the same. In
accordance with this habit the addition of operations is denoted by as well as the addition of numbers. Accordingly we can write where the middle on the left-hand side denotes the addition of the operations and . But, furthermore, we need not be so very pedantic in our symbolism, except in the rare instances when we are directly tracing meanings; thus we always drop the first of a line and the brackets, and never write two signs running. So the above equation becomes which we interpret as simple numerical addition, or as the more elaborate addition of operations which is fully expressed in the previous way of writing the equation, or lastly as expressing the result of applying the operation to the number and obtaining the number . Any interpretation which is possible is always correct. But the only interpretation which is always possible, under certain conditions, is that of operations. The other interpretations often give nonsensical results.
This leads us at once to a question, which must have been rising insistently in the
reader's mind: What is the use of all this elaboration? At this point our friend, the practical man, will surely step in and insist on sweeping away all these silly cobwebs of the brain. The answer is that what the mathematician is seeking is Generality. This is an