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

288 MATHEMATICAL PHILOSOPHY
growing continually larger, the other continually smaller,
though they can never meet, yet they, like the old V's
come to differ by less than any preassigned positive
amount, however small. Undoubtedly the/ new V's, like
the old ones, seem thus to close in upon a common limit.
Do they do so in fact? If they do, what is the limit?
If there be one, it must, we are sure, be something having
a square and having 2 for the square; but this something,
if it exist, can not be one of the things which we and
our race have hitherto meant by number, for, by hypothesis, the only numbers we know in our present stage of
evolution are cardinals, integers and ordinary fractions,
and none of these has 2 for its square. You sense vividly,
I trust, the painful situation into which our limit idea has
brought us. Do you know how we will behave under the
circumstances? How we will try to escape?' By what
means we will endeavor to reach a reconciliation? We
are to suppose ourselves to be dealing with the difficulty
as the mathematicians have dealt with it. Accordingly,
we will not all of us behave in the same way-some of
us will resort to one means of extrication and some to
another. (A) Some of Us will say: V and V' have not
a common S limit, but they have a common S' limit where
S' is a sequence of things we have not yet learned, but
must learn, to recognize and handle; this common limit,
though not a number in the accepted sense of the term, is
something we must regard as having a square and as
having 2 for its square; we will denote the thing by the
symbol, V/2, and call it a number of a new kind-an
irrational number to distinguish it from the old familiar
ratios to be henceforth called rational. (B) Others of us
will say: V and V' have no common limit of any kind
familiar or unfamiliar; it is, however, manifest that they