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

NON-GEOMETRIC INTERPRETATION 87
going geometries, it deals with non-spatial things and so
has a non-spatial content. Some years ago I asked Mr.
Wellington Koo, then a student at Columbia University
and a pupil of mine, a brilliant pupil, in analytical geometry, to tell me what the Chinese word for geometry means
as a word. He replied: "It means show by a figure."
In the interpretation we are about to study we can have
no figures, for figures are spatial affairs. This necessity
of getting on without figures is, in a sense, fortunatefortunate as an intellectual discipline-for, in the absence
of sensuous representation by figures, we shall be driven
to a kind of sheer thinking. And this warning, I hope,
will prepare you for the needed effort.
As in the previous lecture, I will deal first with HAF.
At a later stage of our course, the nature of what is called
the system of real numbers may be discussed. But for the
purposes of the present lecture, I shall assume that you
are sufficiently acquainted with the system, merely reminding you that it is composed of the positive and negative
integers; the ordinary fractions; the irrationals, such
as <z, </7; and the transcendental numbers, like e and ir,
for example. By the term number I shall mean a real
number. In order to indicate the nature and the field
of our new interpretation, it will be convenient to make
use of this definition: If a, b, c be three numbers, b will be
said to be between a and c (or c and a) when and only when
a>b>c or a<b<c, where >means greater than and<
means less than.
The new field of operation-which may be denoted by
N-consists of all dyads (x, y) of real numbers; that is,
of all ordered pairs (x, y), where by ordered I mean that
(x, y) will not be the same as (y, x) unless x=y. It is, of
course, understood that the dyads (x1, yi) and (x2, y2)