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

MORE ABOUT LIMITS 293
into the presence of baffling phenomena, waking our
wonder, giving us a painful sense of failing to see something we ought to see, a sense oflogical suffocation, ofbeing hampered, hemmed in; we seek emancipation and at
length achieve it, not solely by purely logical means, but
partly by observation (as in the case of the segments),
partly by reasoning and partly by an act of will-in short,
by generalization; this deed gives us a new domain of
thought-a new field of ideas (as, for example, the domain
of real numbers); the new domain, once thus established,
is as actual for us as the old one; with reference, however, to any viewpoint within the old one, the new domain
is and forever remains a sheer ideal, not to be attained by
any process or operation-however oft repeated, swift or
prolonged-within the old domain; and finally, a new
domain (as that of the real numbers, for example) may
in its turn become, in the manner indicated, an old one
in relation to another domain (as, for example, the domain
of the complex numbers) which, though itself actual, is,
with respect to the former, an eternal ideal.
Mathematical Limit Processes Fiewed as Species of
Idealization.-In nearing the close of this second long
lecture on variables and limits we come now to what I
most desire to signalize as being for students of philosophy
the most significant aspect of the whole matter. It is this:
In mathematics the great rôle of what we there call limits
and limit processes is in kind identical with the momentous
rôle of that which in other fields of interest we call ideals
and idealization. In the light of the foregoing discussion
the fact is evident, and it shows us again very clearly
what we have repeatedly seen in other connections-that
-far from being detached from common life or alien
thereto,-mathematics is a refined model or prototype