An introduction to mathematics, by A. N. Whitehead.

80 INTRODUCTION TO MATHEMATICS
the statement 2+3 =5. We add 3 to 2 and
obtain 5. Think of the operation of adding
3: let this be denoted by +3. Again 4- 3
= 1. Think of the operation of subtracting
3: let this be denoted by -3. Thus instead
of considering the real numbers in themselves,
we consider the operations of adding or subtracting them: instead of </2, we consider
+/V2 and -V/2, namely the operations of
adding V2 and of subtracting /V2. 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
2+3+1= 2+1+ 3;
so that the addition of the steps +3 and +1
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