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

234 MATHEMATICAL PHILOSOPHY
etry designed for high schools. In high school you probably learned something of the lingo of limits; if you really
there grasped the ideas involved, you were extraordinary
pupils or were very fortunate in your teachers or both.
I say this because in collegiate freshman classes I have
met many students who in their preparation for college
had been exposed to the notion and method of limits, and
I have the impression that among them there were very
few or none whose wisdom in the matter was appreciably
more than phraseological; in the case of most it was even
less. The explanation is not far to seek: the concept of
limit is a subtle concept and the right use of it in mathematical argumentation is a delicate process; these two
things can not be caught, so to speak, on the fly; they
require to be reflected upon again and again; they are
among the things that require to be pondered; but such
meditation, such deliberation upon elusive scientific
abstractions, is one of the things which boys and girls
of the indicated age will not do; it is not their fault;
if they did it they would not be boys and girls; they
would be seasoned logicians and philosophers. If such
understanding of the nature and significance of limits
as you may be supposed to have acquired in high school
has not been deepened and refined by subsequent study,
the intervening years have probably so dimmed your
impressions of the matter that you are now fortunately
able to approach the subject afresh, bringing to bear upon
it critical power of sufficient maturity.
In endeavoring. to analyze the concept of limit it
becomes immediately evident that it involves thé notion
of variable and the notion of constant, together with such
other notions as that of variable and that of constant
themselves involve. What do mathematicians mean by