. It has been pointed out by Strachey [?I that many mathemati-cians treat functions as 'second class objects',, denying them the full generality which is accorded to variables. This attitude is found right at the beginning of algebraic teaching. For example, in an expression like (x+y)^(x-y) it is understood that x and y may vary but, of course, +, x and- may not. Later, in the notation f(x), the student tends to think of x as the variable, and f as a 'constantt function. Similarly, in ALGOL we can write but not (if x> 1 then a else b) + 6 (if x> I. then sin else ~os)(x). Now when we come to consider the problem of defining functions in all their generality it is necessary to be rid of these mental restrictions. We shall mark our emancipation by writing fx instead of f(x), and understand that f is some element drawn from a set of