mathematical constructivism See PHILOSOPHY OF. MATHEMATIC. mathematical function, an operation that, when applied to an entity (set of entities) called its argument(s), yields an entity known as the value of the function for that argument(s). This operation can be expressed by a functional equation of the form y % f(x) such that a variable y is said to be a function of a variable x if corresponding to each value of x there is one and only one value of y. The x is called the independent variable (or argument of the function) and the y the dependent variable (or value of the function). (Some definitions consider the relation to be the function, not the dependent variable, and some definitions permit more than one value of y to correspond to a given value of x, as in x2 ! y2 % 4.) More abstractly, a function can be considered to be simply a special kind of relation (set of ordered pairs) that to any element in its domain relates exactly one element in its range. Such a function is said to be a one-to-one correspondence if and only if the set {x,y} elements of S and {z,y} elements of S jointly imply x % z. Consider, e.g., the function {(1,1), (2,4), (3,9), (4,16), (5,25), (6,36)}, each of whose members is of the form (x,x2) – the squaring function. Or consider the function {(0,1), (1,0)} – which we can call the negation function. In contrast, consider the function for exclusive alternation (as in you may have a beer or glass of wine, but not both). It is not a one-to-one correspondence. For, 0 is the value of (0,1) and of (1,0), and 1 is the value of (0,0) and of (1,1).
If we think of a function as defined on the natural numbers – functions from Nn to N for various n (most commonly n % 1 or 2) – a partial function is a function from Nn to N whose domain is not necessarily the whole of Nn (e.g., not defined for all of the natural numbers). A total function from Nn to N is a function whose domain is the whole of Nn (e.g., all of the natural numbers). See also FUNCTIONALISM , TELEOLOG. F.A.