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

LECTURE XIV
More About Limits
FURTHER DEFINITIONS OF LIMIT-LIMITS AND THE INFINITESIMAL CALCULUS-CONNECTION WITH ORDER, SERIES
AND SEQUENCES-LIMITS AND LIMIT PROCESSES OMNIPRESENT AS IDEALS AND IDEALIZATION IN ALL THOUGHT
AND HUMAN ASPIRATION-IDEALS THE FLINT OF
REALITY-GENIUS AND GENERALIZATION.
THERE are two additional definitions of the term
"limit " with which it is, I believe, very important for
philosophical students to get well acquainted. Both of
them are closely, indeed essentially, connected with what
mathematicians variously call a linear order or a serial'
relation or a series or a sequence. Before presenting them
we must recall clearly to mind some matters briefly
explained in Lecture X and then join therewith certain
kindred ideas and distinctions. You will recall that a
propositional function, say p(x, y), containing two
variables, is said to determine a (dyadic) relation; that,
if 4p(x1, y2) is a true proposition, then and only then the
pair or couple (xi, y2) is called a constituent or element of
the relation; that the class of all such constituents,-the
class of all the pairs verifying (satisfying) 4(x, y), is the
relation; that, if we denote the relation by R, we say
"x has the relation R to y" by writing xRy; that R
accordingly has a sense-so that, if (xi, yi) be a con265