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

THE PSYCHOLOGY OF MATHEMATICS 407
it strictly applied, the very possibility of mathematics,
rightly understood, would be thereby excluded.
It would not be difficult to produce much more evidence of similar kind. But I will content myself with
one further citation-one that is less than a century old
and comes from a mathematician of great power. I refer to Môbius, author of Der barycentrische Calcul
(I827). He saw indeed that, if there were a space of
four dimensions, it would be possible to rotate in it a solid
figure of ordinary space just as a plane figure can be
rotated in the latter space; and he saw that, if such a
rotation of solids were possible, we could make two symmetric solids coincide just as we can make two symmetric
plane figures coincide by rotation in space immersing the
plane. This is perfectly good mathematics, which, how-;ever, he rejects because of a false psychology. His
statement is this: "Da aber ein solcher Raum nicht
gedacht werden kann, so ist auch die Coincidenz in diesem
Falle unmôglich." He meant that such a space cannot
be imagined-he could not have meant that it cannot be
conceived, for he had already conceived it; his blunder
was not one in logic; it was a blunder in psychology-the
psychology of mathematics; though an able mathematician, he did not know that a conceivable space and a conceivable rotation are perfectly good mathematically, even
though they transcend the domain of imagination.
The foregoing facts show clearly that a backward
psychology of mathematics not only operated to hamper
the progress of algebra, but actually delayed, for more
than two thousand years, the advent of the concept of
hyperspace and n-dimensional geometry.
If you turn to the genesis of non-Euclidean geometry,
you find an essentially similar tale. The birth was baf