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

162 MATHEMATICAL PHILOSOPHY
words (synonyms). A telephone directory is a similar
example, more nearly mathematical than the other one.
Indeed, you will find, by a little looking about, that such
illustrations abound on every hand. A perfect example of
a genuinely mathematical one-to-one transformation running both ways is afforded by the vulgar process of counting
the objects of a class, a class of count-words (one, two,...)
being transformed, in a certain order, into the class (of
objects to be counted), and conversely.
A moment ago I told you that, when a boy, my second
puny introduction to the concept of mathematical transformation occurred in a dry little chapter (called transformation of coordinates) located near the middle of a
beginners' course in analytical geometry. You will recall
that in a previous lecture of the present course I gave a
very brief introduction to the analytical geometry of the
plane. As you will recall, it was shown how to transform
(in one-to-one fashion) the class of the plane's points into
the class of the real number pairs (x, y), and, conversely,
the latter class into the former; we saw that the former
transformation gives birth to the method of analytical
geometry; and the converse transformation, to the converse method-that of geometric analysis. We may
think of the two transformations as one having two
converse aspects, and, following usage, may speak of the
two methods as one-called analytical geometry. Now
observe carefully that the analytical geometry (of the
plane), instead of merely using transformations among
its processes, actually springs out of-owes its very
existence to-a transformation, that of points into number pairs and of such pairs into points. Nay, the whole
of analytical geometry viewed (properly) as a method is
simply a vast transformation based upon the one just