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

158 MATHEMATICAL PHILOSOPHY
simple law which requires us to associate, in thought, each
element of C with its double; symbolically expressed the
law is: ( -> 2), (2 ->4), (3 -> 6),..., (n - 2n),...;
you observe that the class C is thus transformed into the
class C' by a one-to-one transformation; now, C' is, as
said, a part of C, so that C has been, you see, transformed,
by a one-to-one transformation, into a part of itself; and
that is remarkable, for it can not, as you know, be done
with just any class-it can not, for example, be done with
a class composed of one thing, or two, or three, or a dozen,
or a dozen million. If a class be such that it can be thus
transformed into some part of itself, it is said to be
infinite-an infinite class. The concept denoted by the
term, " infinite class," is one of the most important of our
modern mathematical and philosophical concepts. In
noticing that it is defined by means of transformation,
you get a glimpse of the latter's fundamental importance.
May I here relate a bit of relevant personal experience?
Some years ago a student of philosophy and I undertook
to read together a pioneer work in the mathematical
doctrine of infinity-the Paradoxien des Unendlichen by
Bernhard Bolzano, mathematician, philosopher and theologian. We came to a passage where Bolzano shows
that the class of points composing a straight line segment
is an infinite class. He does it very simply and very
clearly by showing, about as follows, that the segment
FIG. 20.
can be transformed, in a point-to-point fashion, into a
part of itself. Let y=lx; use the equation as a law of
transformation converting a point whose distance from