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

INVARIANCE 185
are two objects of thought, for they are evidently not the
same in all respects. Let us now suppose D transformed
into D', (D > D'), by a transformation T-I do not mean
transmutation, I mean association of D with D', for that,
as we have seen, is at bottom what a mathematical
transformation is, that and nothing more. Among the
variant properties of D under T are certainly time and
place, and possibly weight and distance from the Moon.
What, if any, are the invariants? Subject to some correction by the refinements of modern physics, it is yet instructive to answer that among the invariant properties of
D under T are shape, size, mass, degree of hardness,
capacity for light absorption, and so on. Some of these
will, of course, not be invariant under transformation of
D into D', where D' denotes D crushed.
For an example drawn from a very different field, let
P denote the personality of John Smith at the age of I5,
P' his personality at the age of 30, and let someone, say
Smith himself at the age of 45, transform P into P',
(P -> P'), by a transformation T-again I do not mean
transmutation, the mysterious process of a boy's becoming a man. The variant properties of P under T are
obviously many-years, for example, wisdom, folly, interests, hope, and so on; among invariants are the properties
of being a son, of being a man, of being a human, of being
what Count Korzybski calls a time-binder, of being a
visible object; another one-of extraordinary interestis the property called personal identity. This last property, which runs through a long sequence of personalities,
exemplifies an immense class of important invariants that
no one has been able to formulate precisely though their
existence is manifest: we may call them unformulated
or qualitative invariants. These are not indeed strictly