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

416 MATHEMATICAL PHILOSOPHY
imagination is not. My theme is: The symmetry of
thought and the asymmetry of imagination.'
Consider the simple algebraic expression
lXl +U2x2 +... +UnX + I
The u's precede the x's, but that is here of no importance,
for, owing to the commutative law of ordinary multiplication, ux is equivalent to xu; if we replace the u's by the
corresponding x's and the latter by the former, the
expression remains algebraically unaltered. On that account we say that the expression is symmetric with respect
to the u's and the x's. Such interchange of the u's
and the x's may be likened to the interchange of two
opposite halves of a perfectly symmetric tree-the figure
of the tree as a whole remains unchanged. It will be
convenient to denote the expression by the symbol
E(u, x)-the symbol E(x, u) would, of course, do just
as well but let us use the former.
Now consider the equation
(I) E(u,x)=0
It is, like the expression, symmetric in the sense defined.
We may interpret the equation geometrically. To do so,
let us view the x's as coordinates of a point in a pointspace, Sn, of n dimensions. If we suppose the u's to have
definite values the equation (I) imposes one condition
on the mobility of the point (Xi, x2,..., Xn); and so the
equation represents,-has for its locus, as we èay,-a
space S,,_ of points. If we give the u's another set
of values, thus obtaining a new equation of form (i),
1A paper on this subject which I presented at the Princeton meeting
of the American Philosophical Association (191o) was published in the
Journal of Philosophy, Psychology and Scientific Method, June 8, 9gII.