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

168 nMATHEMATICAL PHILOSOPHY
relation and a transformation; given a relation, you have
a transformation and a function: one thing-three aspects;
and the fact is exceedingly interesting and weighty.
Impressed by the immeasurable scope of the ordinary
function concept, some thinkers have said, with a striking
approximation to truth, that mathematics and indeed
the whole of science is just the study of functions, It
çan, you see, be said, with the same approximation to
truth, that the whole of science, including mathematics,
consists in the study of transformations or in the study
of relations.
Time is lacking for extensive pursuit of the matter
here. Before leaving it, however, I should like to sig
nalize the parallelism in another way. A relation R
has what is called a domain,-the class of all the terms
such that each of them has the relation to something or
other,-and also a codomzain-the class of all the terms
such that, given any one of them, something has the
relation to it; a transformation T proceeds from a class,that of the things transformed,-to a class-that of the
transforms; the independent variable (or argument, as
it is called) of afunction F has a range,-the class of values
the argument may take,-and the function has a range,the class of values the function may take. Note the
matching of the foregoing things; it is easiest to do it by
an example. Consider the simple propositional function:
x-zy, It determines a relation R, a transformation T
and a function F (i.e., x, or 2y). Let K denote the class
of real numbers and K' the class of their doubles. You
see that K is at once the codomain of R, the class transformed by T, and the range of F's argument y; also that
K' is at once the domain of R, the class of transforms
(under T), and the range of F. In this particular example