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

HYPERSPACES 323
thus designated are either more or less genuine than other
numbers.
The system of real numbers is a vast system or classthe multitude of the numbers in it is indeed very great.
The extent of the multiplicity,-the number, if you please
of all the real numbers,-is often conveniently denoted by
the familiar symbol for infinity, oo. The same symbol
is used to indicate how many things, elements or members
there are in any other equally numerous system or classsay, that of the points in a straight line. If x be a variable representing " any one " of the real numbers, we say
that x has one degree of freedom; we say also that the
system of real numbers is a one-dimensional system or
class; in like manner we say that a point P, if free to
move along a straight line, has, in virtue of that fact, one
degree of freedom, and that a line, regarded as a system
of points, has one dimension or is one-dimensional.
Now, a real number is one thing and a pair of them is
another. Such pairs constitute a system (of pairs). How
many pairs are in the system? It is easy to tell. Let
the symbol (x, y) be a variable representing any one of
the pairs; give y some definite value, say, yl, and let x
vary-x can take oo values; each of these taken with yi
gives a pair, and so we get oo pairs; in each of these we
may replace yi by any of the oo of values that y may take;
we so obtain in all, as you see, oo times oo, or oo 2, pairs.
It is plain that the variable (x, y), an arbitrary or undetermined pair of the system, has two degrees of freedom,
owing to the variability of the two parameters, or coordinates, x and y. Thus the system of all the pairs of real
numbers is a two-dimensional system (of pairs). You see
at once that the system of triads or triplets of real numbers
has three dimensions, or is a tri-dimensional system (of