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

292 MATHEMATICAL PHILOSOPHY
mathematics, but also and especially in understanding the
ways of spiritual life-the ways of truth to men.
Not to wish to dwell in this insight long enough to
make it our own would show us unworthy-stupid or
perverse. We might indeed illustrate it in many ways and
in many connections. We might show in detail how limit
concepts at work in the domain of real numbers, especially
in connection with equations failing to have roots in that
domain when the variable coefficients are allowed to
approach certain limits, make us keenly aware that the
domain of reals, vast as it is, is yet too meagre for our
purposes, and how we are thus led to effect another
immense number-generalization-that one, I mean, which
gave us what we call the complex numbers (x+iy, x and y
being reals, and i being the so-called imaginary unit,
~/-I), now the subject of a stately theory having wide
application in physics and even in engineering; we might
show in detail, little step by step, how limit concepts at
work in geometry have availed so to extend or generalize
such fundamental notions as length, area and volumeformerly clear and well defined only in connection with
broken lines, or polygons, and solids bounded by planesthat we can now confidently and understandingly use the
notions in connection with all manner of curves and
curved surfaces. But such details would require certainly
more time and perhaps more patience than we now have
at our disposal.
I must, however, once more insist upon the matter
which I mentioned a moment ago and which I have emphasized elsewhere. The matter is this: a limit-begotten
generalization always originates in the work of some limit
concept operating in some established domain (such as that
of our ratios, for example) wherein the concept leads us