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

Se2 MATHEMATICAL PHILOSOPHY
space," where n is greater than three? I have said "the
meaning" as if there were only one. As a matter of fact,
the term has three meanings, which, though they are
closely related, it is essential to distinguish if we are to
avoid confusion. I shall try to explain them clearly, reserving the most interesting one for the last. Certain
refinements of refinements I shall avoid as likely to obscure and hinder rather than to clarify and help in a first
presentation. It will be very advantageous as a preliminary to speak of some simple matters connected with
the system of real numbers.
There is, as you doubtless know, an extensive and very
refined theory of the logical genesis and properties of these
numbers. The theory is to be found in the numerous
works dealing with functions of a real variable, the
profoundest treatment of the subject being that in the
Principia Mathematica. I am not going to assume that
you are familiar with that theory. I take it for granted,
however, that you are sufficiently acquainted with the
system of real numbers to understand fairly well what I
purpose to say about it here. You are aware that the
system is composed of the infinitude of positive and
negative integers, the infinitude of rational fractions, and
the infinitude of irrational numbers like x/2, for example,
and including the so-called transcendental numbers such,
for example, as the familiar specimens, ir and e. Historically these numbers were called " real " to distinguish
them from the so-called "imaginary" numbers, like
/- 2, for example, which latter were long regarded, quite
unjustly, as an ungenuine kind of number. Today the
old adjectives, real and imaginary, are still regularly
employed, but they no longer signify that the numbers