Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

TRANSFORMATION 167
concept of mathematical transformation, the concept of
relation and the (ordinary) concept of mathematical
function are, as I have said, virtually but three aspects
of one and the same thing seen from different points of
view. For look again at the propositional functions:
(I) X=i(I —3y), (I') y=(I-2x). Observe that (I) is:
(a) a law of transformation by which a class of numbers y
is converted into a class of numbers x; (b) a determiner
of a relation, namely, that composed of the couples
(x, y) verifying (I); (c) a determiner of x as a function
(in ordinary sense) of y, namely, the function 2(I-3y).
Observe that (I') determines at once the converse transformation, the converse relation and the converse,-commonly called the inverse,-function. Observe that if
xi and yi verify (I), then the pair (xi, yi) is: (d) composed
of a thing transformed and its transform; (e) a pair of
values of the function; (f) a couple, or constituent, of the
relation. Of course, the like is true of (i').
Look again at the propositional functions: (2) x is a
parent of y; (2') y is a child of x. You see that (2) is at
once: (g) the determiner, or law, of a transformation, associating any given x (a parent) with some y or y's (child or
children of the x), the y or y's being the x's transform or
transforms; (h) the determiner of x as a function (in
ordinary sense) of y, for to any value of y (some child)
there corresponds a value or values of x (some parents);
(i) the determiner of a relation, composed of the couples
of verifiers of (2). It is plain that (2') yields the respective converses of the foregoing transformation, function
and relation.
The connections shown by these particular examples
hold in general: given a transformation, you have a
function and a relation; given a function, you have a