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

!21 MATHEMATICAL PHILOSOPHY
into Q', the end of the J' beginning at Q; now there is a
vector beginning at P and running direct to Q'; and so
there is a transformation T" converting P into Q'; it is
this T" that we shall mean by ToT'. Without further
talk, you see that our group of angel flights, or vectors,
now appears as an infinite Abelian group of transformations
(of our space of points into itself). Such transformations
do not involve motion in fact; it is customary, however,
for mathematicians to call them motions, or translations,
of space; 1, for example, being thought of and spoken
of as a translation of the whole of space (as a rigid body)
in the direction and sense of V and through a distance
equal to P's length. In accordance with this stimulating,
albeit purely figurative, way of speaking, the group in
question is the group of the translations of our space.
We are now in a good position to glimpse the very
intimate connection between the idea of group and the
idea of invariance. Suppose we are given a group of
transformations; one of the big questions to be asked
regarding it is this: what things remain unaltered,remain invariant,-under each and all the transformations
of the group? In other words, what property or properties are common to the objects transformed and their
transforms? Well, we have now before us a certain group
of space transformations-the group of translations.
Denote it by G. Each translation in G converts (transforms, carries, moves) any point into a point, and so
converts any configuration F of points,-any geometric
figure,-into some configuration F'. What remains unchanged? What are the invariants? It is obvious that
one of the invariants,-a very important one,-is distance;
that is, if P and Q be any two points and if their transforms under some translation be respectively P' and Q',