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

THE MATHEMATICS OF PSYCHOLOGY S93
"Grand Continuum" as it was called by Professor
Sylvester. It contains such symbols as a/2, e, r and
countless hosts of others sandwiched between the symbols
constituting the first-order continuum, much as the
rational fractions of this are sandwiched between the
cardinal numbers.
The Grand Continuum has been a subject of profound
investigation by mathematicians, especially during the
last half-century, and has long been everywhere an
important theme of university instruction in what is called
the theory of the real variable. This instruction, which
has found its way into numerous text-books on function
theory, is mainly based, directly or indirectly, upon three
classical expositions of the matter. I refer to Dedekind's
exposition, which has been translated by Beman and
Smith and with another of the author's works has been
published under the title Essays on Number; to that by
Georg Cantor in his creative memoirs on Mannigfaltigkeitslehre (or Mengenlehre); and to the exposition found
in the works of Weierstrass. For our present purpose it
will be sufficient to remind ourselves briefly of one way
in which the concept of the Grand Continuum may be
formed and of its two essential or definitive marks. Consider the following two sequences of rational numbers:
first, the sequence of all rational numbers such that each
of them is less than 2; second, the sequence of all rational
numbers such that the square of each of them is less than 2.
Each of the sequences approaches, as we say, a definite
somewhat as a limit. The limit of the first is 2, which is
rational; the limit of the second is not rational; we call
it irrational, denote it by the symbol V/2, and say that this
irrational is given or defined by the sequence (or any
other sequence) having it for limit. There are infinitely