ertain words, 'one,' 'two,' 'three,' etc., in
the mind. Many savage peoples have discovered no better method of counting
than by setting up a one-to-one correspondence between the objects to be
counted and their fingers. The scientist who busies himself with naming
and classifying the objects of nature is only setting up a one-to-one
correspondence between the objects and certain words which serve, not as a
means of counting the objects, but of listing them in a convenient way.
Thus he may be able to marshal and array his material in such a way as to
bring to light relations that may exist between the objects themselves.
Indeed, the whole notion of language springs from this idea of one-to-one
correspondence.

*2. Consequences of one-to-one correspondence.* The most useful and
interesting problem that may arise in connection with any one-to-one
correspondence is to determine just what relations existing between the
individuals of one assemblage may be carried over to another assemblage in
one-to-one correspondence with it. It is a favorite error to assume that
whatever holds for one set must also hold for the other. Magicians are apt
to assign magic properties to many of the words and symbols which they are
in the habit of using, and scientists are constantly confusing objective
things with the subjective formulas for them. After the physicist has set
up correspondences between physical facts and mathematical formulas, the
"interpretation" of these formulas is his most important and difficult
task.

*3.* In mathematics, effort is constantly being made to set up one-to-one
correspondences between simple notions and more complicated ones, or
between the well-explored fields of research and fields less known. Thus,
by means of the mechanism employed in analytic geometry, algebraic
theorems are made to yield geometric ones, and vice versa. In geometry we
get at the properties of the conic sections by means of the properties of
the straight line, and cubic surfaces are studied by means of the plane.

[Figure 1]

FIG. 1

[Figure 2]

FIG. 2

*4. One-to-one correspondence and enumeration.* If a one-to-one
correspondence has been set up between the objects of one set and the
objects of another set, then the inference may usually be drawn that they
have the same number o