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

324 MATHEMATICAL PHILOSOPHY
triads); that the system contains oo3 triads; and that an
undetermined triad (x, y, z) enjoys three degrees of freedom, one for each of the mutually independent coordinates,
or parameters, x, y and z. The generalization is obvious
and easy; if we think of the system of all the sets
(Xi, X2,... Xn) of real numbers, each set composd of n
numbers, it is readily evident that the system is an
n-dimensional system (of sets); that it contains ooe sets;
and that a free, or undetermined, set of the system
possesses n degrees of freedom.
I have been speaking mainly of numerical things. I
am now going to speak of things geometrical or spatial,
and will first make a little more precise what I said a
moment ago regarding a line. Let L be a straight line;
choose a point of L for origin of distances and mark it O;
let P denote any point of L; denote by x the distance (in
terms of some chosen unit) from O to P, it being understood that x is positive or negative according as P is on the
one side of O or on the other-of course x will be zero if
P coincide with 0. Thus a reciprocal one-to-one correspondence is set up between the real numbers of the
system thereof and the points of L; x represents P numerically, P represents x geometrically; and x is the coordinate or parameter of P. Notice that L is here the field
or the space of operation and that we are regarding it as a
field or a space of points. In the light of the preliminary
discussion respecting numbers, you see immediately that
a straight line, regarded as a space of points, is onedimensional, since it perfectly matches, as indicated, the
one-dimensional system of the real numbers. In other
words a line is a point-space of one dimension. You
catch the idea:: if a point of a line depended upon two or
more coordinates instead of only one, we should say that a