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

TRANSFORMATION 155
of mathematical science. For, while this science is, as
we have seen, composed of doctrinal functions, these
forms are themselves woven of abstract, or formal, relations.
In this lecture, I purpose to deal with the mathematical idea denoted by the term "transformation." I need
not say that an hour's lecture can neither impart much
knowledge of transformation theory nor give skill in the
use of transformations as instruments; for the former
requires prolonged study and the latter is the slow-maturing fruit of practice. What I hope to do in the hour is
to make clear what mathematicians mean by a trans.
formation and by a law of transformation; to show how
fundamental and omnipresent the process of transformation is in all our thinking; to give an inkling of the endless number and endless variety of existing transformations; to show how transformations appear now as power.
ful tools and now as interesting themes; to disclose the
intimate connection of the notion of transformation with
that of relation and that of function; and briefly to indi.
cate how the phenomena of transformation in the abstract static world of mathematics correspond to the
phenomena of change in the concrete dynamic world of
sense.
Let me begin as simply as I can-so simply, indeed,
as possibly to suggest that transformation is a trivial
term, which it is infinitely far from being. The notion of
transformation has its root in the power we have, when
given any two objects of thought, to associate either of
them with the other. If a and a' be two such objects, we
can, in thought, associate: (i) a with a'(a ->a'); or,
conversely, (2) a' with a(a - a'); or (3) each with the
other (a <-> a'). If we do (I), we say we have trans