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

INVARIANCE 183
has the peculiar properties pl, p2...; that a' has the
peculiar properties pi', p2',...; and that the properties
7ri, r2,... belong to both a and a'; we say that the properties wri, 2,..., since they belong both to a and to its
transform a', are invariant under T,-have suffered no
change,-are preserved; and that pl, p2..., as they
belong to a but not to its transform, are variant under T,properties lost under the transformation,-not carried
over by it. It is plain that under the converse (commonly
called the inverse) transformation T', (a <- a'), Pi' p2',...
are variant while,7r, r2,... are invariant as before. It is
evident that, if a property be invariant under some
transformation, it will be invariant under the converse
transformation. I am aware that what I have now said
is so general, abstract and simple as to make the concept
dealt with seem unreal-tasteless, pallid, thin, intangible.
But the seeming is seeming only. The idea in question,
far from being detached from reality, literally pervades it
-pervades our thinking about it and our handling of it.
How may we convince ourselves that this is true? We
may do it by looking about us a little and by a little reflection-by considering a few specific concrete examples
and observing that such examples abound in countless
multitudes on every hand.
For a few examples that everyone can understand,
consider the following. Let I denote the class of familiar
integers:, 2, 3,...; and suppose these to be transformed in accordance with the law: (I) y = 2x. The transforms constitute the class of even integers: 2, 4, 6,....
We note that integers are converted into integers, and so
the property of being an integer is preserved-it is an
invariant under transformation (I); the value of an
integer, however, is a property not preserved-it is