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

VARIABLES AND LIMITS 255
terms not in it in respect to the same kind k. Examples
abound, and need not be cited. If, with respect to a given
kind of difference, two terms be identical, we shall say
that the amount of their difference (of the given kind)
is null or naught; if, as often happens, the given kind of
difference be numerical, then the hypothetical identity
is numerical equality and we shall say, in accordance with
usage, that the amount of difference is zero; we shall thus
be using zero as a special variety of the foregoing null or
naught. Here, as elsewhere, it is essential to use good
sense. It would, for example, be nonsense to speak of
two numbers as differing in respect to color or patriotism
or loyalty. When we speak of two terms as having an
amount of difference of a kind k, it is to be understood that
k is a kind with respect to which the terms can be significantly compared. A kind k of difference may be
called a k-difference; and a given amount d of it, the
k-difference d.
I am now going to define a very convenient idea to be
called a k-neighborhood of a term t. Let t be a term
comparable with one or more terms in respect to a kind k
of difference, and let d be a given amount (greater than
null) of such difference; then d is said to determine a
k-neighborhood of t. If d grow larger or smaller, the
neighborhood will do likewise. If d be specified, we may
speak of the k-neighborhood d of t. Observe that d
determines the same neighborhood for each and every term
comparable with one or more terms in respect to the
given difference-kind k. A term t' will be said to be in
or not in the k-neighborhood d of t according as t' differs
from t by an amount less than or not less than d. You
see at once that, if t' be in the k-neighborhood d of t, t is
in the k-neighborhood d of t'. Obviously t is itself in all