Peano's The Principles of Arithmetic 1889

Statement

Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision. First Peano defines the sign "K means class, or aggregate of objects", the objects of which satisfy three simple equality-conditions, a = a, (a = b) = (b = a), IF ((a = b) AND (b = c)) THEN (a = c). He then introduces φ, "a sign or an aggregate of signs such that if x is an object of the class s, the expression φx denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects φx; secondly, that "if x and y are objects of class s and if x = y, we assume it is possible to deduce φx = φy". Given all these conditions are met, φ is a "function presign". Likewise he identifies a "function postsign". For example if φ is the function presign a+, then φx yields a+x, or if φ is the function postsign +a then xφ yields x+a.