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

THE PSYCHOLOGY OF MATHEMATICS 421
lien of the elements,-lineoids,-of which it is the envelope.
And now the fact to be noticed is this: I(u) images elements, but not their bond; I(x) images the bond, but
not the elements.
It is plain, too, that I(u) is more satisfactory than
I(x). This fact becomes obtrusively evident if we
geometrize T(u) and T(x) themselves. The two geometries,-which we must remember are conceptual,-match
each other in fact-to-fact fashion perfectly; with respect
to each other they are perfectly symmetric. In the two
geometries a point of T(u) corresponds to a lineoid of
T(x), and a line-segment joining two points of T(u)
corresponds to the angle of two lineoids of T(x). Now
it is evident that the image of a segment is very superior
to any image we can form for the angle between two
intersecting lineoids. I need not give further examples,
which are endless in number and tell the same tale. If
you desire to do so, you can pursue the matter in S4
and in higher and higher spaces.
The conclusion is that, in relation to space, conception
or thought is perfectly symmetric and that imagination or
intuition is asymmetric. As n increases, thought continues to look about in spaces of ever-ascending dimensionality like a binocular being with no impairment of its
twofold vision; its light is spread abroad equally everywhere; whilst imagination's eyes not only fail more and
more as n mounts higher, but they fail in unequal measure.
To change the figure, thought enters and moves about
freely in the hyperspaces like an eagle with both wings
equally outspread and always adequate for any zone
however vast or high, but the movement of imagination
there is like the flight of a bird of feeble and failing wings,
unable to rise and soar.