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

VARIABLES AND LIMITS 253
Note that we have here an endless (downward running)
row (S), of endless rows of polygons. It is obviously natural to denote by K-P, any one-the nth one-of the
rows in the row (S). The symbol K-Pn thus becomes
a variable whose range is, not (S), but the class of rows
in (S). Our new variable is plainly not numerical;
neither is it geometric for, though the rows,-the terms in
its range,-are composed of polygons, a row of geometric
figures is, in strictness, no more a geometric figure than a
row of men is a man.
For an example of a variable more evidently, though
not more actually, non-geometric and non-numerical than
the preceding one, we may take p where p represents any
one of the propositions derivable from some given propositional function p(x) by means of its verifiers. The
range of p is obviously the class of true propositions
having 0(x) for matrix.
Conceptions of Limit.-It is evident that much of our
human thinking,-I strongly suspect that all of it,-is
concerned with variables. One of the lessons which the
history of thought and our personal experience in thinking
teach very clearly is that we can not deal with variables
logically or with any close approximation to rigor without
the help of the notion (or notions) which mathematicians
denote by the term " limit." We must, accordingly, try
to understand what the term means. I have just now
used the plural-conceptions of limit-instead of the
singular. I have done so because there are various
meanings of the term and I intend to present more than
one of them. The definitions I am going to present are
closely related, but they are not equivalent: they differ
in content and scope, and you will find it very instructive
to compare them in these respects. You will observe at