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

156 MATHEMATICAL PHILOSOPHY
formed (or converted) a into a' and that a' is the transform
of a; if we do (2), we transform a' into a, which is then
the transform of a'; if we do (3), we transform each into
the other and each is the other's transform.
Let us take another step: suppose C is a class composed
of a, b, c, and that C' is a class composed of a', b', c'. We
may transform in the manner,-according to the law,shown in the table (I): (a -> a'), (b -> b'), (c -> c'); or by
the law (2): (a -> a'), b - c'), (c - b'); or by the law
(3): (a ->b'), (b a'), (c c'); or by the law (4):
(a - b'), (b - c'), (c -* a'); or by the law (5): (a - c'),
(b - a'), (c - b'); or by the law (6): (a -- c'), (b - c'),
(c -> a'). Ir each case we have, we say, transformed the
class C into the class C', and C', we say, is the transform
of C'. You see, too, that we could conversely transform
C' into C in six corresponding ways-by any one of six
corresponding laws, which we could express by tables as
above. You see, too, that we could transform the classes
C and C' each into the other, now by one law, now by
another, as, for example by the law (7): (a < a'),
(b -> b'), (c (<- c'); or by (8): (a — > b'), (b <- a'),
(c *-> c'); and so on. Notice that in each of the transformations (I),..., (6), each element of C has only one
transform and that each element of C' is a transform;
each of those transformations is a one-to-one transformation, and has a direction, or sense, namely, from C to C';
not from C' to C; observe that in (7) or (8) the transformation is again one-to-one but runs both ways. In each of
the cases considered, one of the classes is transformed into
the whole of the other, not merely into a part of it. But
such need not be so; we can transform two or more
elements of C into a same one of C' or one of C into more
than one of C' and thus transform C into a part, or sub