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

252 MATHEMATICAL PHILOSOPHY
What kind? What are their ranges? Are these finite
or infinite? Do the functions increase or decrease with n?
All increase with n except Pn; how justify the exception?
Can n so increase that Pn will coincide with K? Or Ln
with L? Or An with A? Or dn with R? Why not?
the differences L-L,, A -An, R-dn are variables, functions of n-are they not? If I name a positive number ~
as small as I please, can you then choose such a value for
n,-such a term in n's range,-that the foregoing differences will be less than o? Having thus chosen n, if you
let n take still larger and larger values, will the differences
in question keep always less than v? Will they ever
vanish-that is, be equal to zero? Why or why not?
(i ) Let us now consider two variables that are
neither geometric nor numerical. Referring to Fig. 26,
let us write K-P1, K-P2,...; these seem to be symbols
but what, pray, do they symbolize? They are indeed
suggestive, but they have at present no definite meanings,
for neither K nor any of the P's is a number, K being just
a circle-a specific circle-and each P a specific polygon.
We may, however, assign meanings to the ostensible
symbols, thus making them genuine symbols. What
meanings shall we assign? A little reflection upon the
make-up of Fig. 26 naturally suggests that we let K-Pi,
K-P2, K-P3,..., denote respectively the endless
polygon-rows indicated at the right of them below:
K-P1: P2, P3, P4, P5,...;
K-P2: P3, P4, P5,.....;
(S) K-P3:P4,P5,........;
* * * e * e e