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

182 MATHEMATICAL PHILOSOPIHY
knowledge of it can be gained only by pursuing it through
many months with the tireless energy of a sleuth-hound.
It is my aim to give you just a little introduction to the
matter, a clue to it, a good grasp of its central idea, a very
slight acquaintance with its methods, and a fair sense of
its general significance and its bearings as a prototype
for that quest of abiding reality which has dominated all
the great truth-seeking activities of man and has served
to unite them,-religion, philosophy, art, science,-as but
different aspects of one supreme enterprise: emancipation
from the tyranny of change-discovery of a stable worlda haven of refuge from the raging tempests of the sea.
Let us begin as we began in the case of transformation
-as simply as we can; indeed, to begin aright we must
return and begin our new study just where we began to
study the meaning of mathematical transformation; for
we may say at once that, in general sense, an invariant,
as the word indicates, is to signify something which, when
other things connected with it suffer change, remains
itself unchanged; and now change, as we have seen, is
represented in logic (in mathematics) by means of relations
which, as we have also seen, mathematicians call transformations; so that the mathematical term " invariant"
or " invariance " would be unintelligible or meaningless
save for its connection with the mathematical notion of
transformation.
We will accordingly suppose, as in the preceding lecture, that a and a' denote two objects of thought and that
by a transformation-which we may denote by T-a has
been transformed, or converted, into a', (a -> a'). Now,
an object of thought has what we call properties, some (at
least one) of which are peculiar to it and some belong to
one or more other objects as well. Let us suppose that a