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CHAPTER I — ONE-TO-ONE CORRESPONDENCE

*1. Definition of one-to-one correspondence.*
Given any two sets of individuals, if it is possible
to set up such a correspondence between the two sets
that to any individual in one set corresponds one and
only one individual in the other, then the two sets
are said to be in *one-to-one correspondence*
with each other. This notion, simple as it is,
is of fundamental importance in all branches of science.
The process of counting is nothing but a setting up
of a one-to-one correspondence between the objects
to be counted and certain 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