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

THE PSYCHOLOGY OF MATHEMATICS 417
the new equation will represent another space of n -
dimensions. It is thus plain that the u's serve for coordinates for a variable S._- in S~ just as the x's serve for
coordinates of a point in Sa. Since the u's may take as
many different systems of values as the x's may take, you
see that the space S., in which we are operating, contains
as many S,_1's as it contains points.
We have just now seen that, if the u's be held fixed
in value and the x's be allowed to vary subject to condition (I), this equation represents some definite S,_ as
the ensemble or locus of the points contained in it. Now
note very carefully the reciprocal or dual, as it is called,
of the fact just stated. The dual is that, if the x's be held
fixed in value (thus giving us a fixed point, say, P) and
the u's be allowed to vary subject to condition (I), the
equation represents P as the ensemble or envelope (as it
is called) of all the S,,'s containing it.
Naturally the two interpretations-one for the u's
fixed and the x's variable, the other for the x's fixed and the
u's variable-of one and the same equation
E(u, x) = O = E(x, u)
may be significantly described as symmetric interpretations. Indeed, as you readily see, if in the conceptual
space Sn (of operation) we interchange the notion of
point (as an envelope of S,_,'s) and the notion of Sn,_
(as a locus of points), Sn will as a whole remain, like the
initial expression, like our equation, like our symmetric
tree, absolutely unchanged. Under the mentioned operation, Sn is an invariant. In the same way, systems of
equations like the foregoing one admit of symmetric
interpretations. But I shall not deal with such systems
where n is general. It will be easier for you, and for my