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

THE MATHEMATICS OF PSYCHOLOGY 395
line and the real numbers. Such a continuum of points
or of real numbers is a linear or one-dimensional continuum of second order. The ensemble of pairs of real
numbers or the ensemble of points of a plane (as ordinarily conceived, in analytic geometry, for example) is
a two-dimensional continuum of second order; for a threedimensional one, it suffices to refer to the ensemble of
triplets or triads of real numbers or to the ensemble of the
points of our familiar geometric space as ordinarily conceived. And it is evident that second-order mathematical continua may have any given dimensionality
whatever. For a logically much more refined account of
the system of real numbers, you should examine Russell's
Principles and especially the Principia. I am giving here
but a sketch of the usual account.
The Definitive Marks of a Grand Continuum.-What
are the characteristic or definitive marks or properties of
a mathematical continuum of second order? The answer
is: an ensemble of numbers or points or other elements is
such a continuum when and only when the ensemble is
compendent and perfect. These are technical terms. What
do they mean? Let us answer in terms of points. An
ensemble of points is compendent (or zusammenhangend
as the Germans say or connected as it is common to say
in English) if it be such that, given any two points of it,
it is possible, by stepping only on points of the ensemble,
to pass from one of the given points to the other by a
finite number of steps, where each step is equal to or less
than some previously assigned distance, however small.
An ensemble of points is perfect, provided it be identical
with the ensemble of its limit- points, where, by a limitpoint of an ensemble, is meant a point such that there are
points of the ensemble distant from the given point by an