An Elementary Course in Synthetic Projective Geometry | Page 4

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 of elements. If, however, there is an infinite
number of individuals in each of the two sets, the notion of counting is
necessarily ruled out. It may be possible, nevertheless, to set up a
one-to-one correspondence between the elements of two sets even

when the number is infinite. Thus, it is easy to set up such a
correspondence between the points of a line an inch long and the points
of a line two inches long. For let the lines (Fig. 1) be AB and _A'B'_.
Join _AA'_ and _BB'_, and let these joining lines meet in S. For every
point C on AB a point _C'_ may be found on _A'B'_ by joining C to S
and noting the point _C'_ where CS meets _A'B'_. Similarly, a point C
may be found on AB for any point _C'_ on _A'B'_. The correspondence
is clearly one-to-one, but it would be absurd to infer from this that there
were just as many points on AB as on _A'B'_. In fact, it would be just
as reasonable to infer that there were twice as many points on _A'B'_ as
on AB. For if we bend _A'B'_ into a circle with center at S (Fig. 2), we
see that for every point C on AB there are two points on _A'B'_. Thus it
is seen that the notion of one-to-one correspondence is more extensive
than the notion of counting, and includes the notion of counting only
when applied to finite assemblages.

*5. Correspondence between a part and the whole of an infinite
assemblage.* In the discussion of the last paragraph the remarkable fact
was brought to light that it is sometimes possible to set the elements of
an assemblage into one-to-one correspondence with a part of those
elements. A moment's reflection will convince one that this is never
possible when there is a finite number of elements in the
assemblage.--Indeed, we may take this property as our definition of an
infinite assemblage, and say that an infinite assemblage is one that may
be put into one-to-one correspondence with part of itself. This has the
advantage of being a positive definition, as opposed to the usual
negative definition of an infinite assemblage as one that cannot be
counted.

*6. Infinitely distant point.* We have illustrated above a simple method
of setting the points of two lines into one-to-one correspondence. The
same illustration will serve also to show how it is possible to set the
points on a line into one-to-one correspondence with the lines through a
point. Thus, for any point C on the line AB there is a line SC through S.
We must assume the line AB extended indefinitely in both directions,

however, if we are to have a point on it for every line through _S_; and
even with this extension there is one line through S, according to
Euclid's postulate, which does not meet the line AB and which therefore
has no point on AB to correspond to it. In order to smooth out this
discrepancy we are accustomed to assume the existence of an infinitely
distant point on the line AB and to assign this point as the
corresponding point of the exceptional line of S. With this
understanding, then, we may say that