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

406 MATHEMATICAL PHILOSOPHY
evident that the psychology of mathematics had not yet
learned to discriminate the conceivable from the imaginable. We now know that in a space of n dimensions n
lines can be mutually perpendicular and we know that
such spaces are geometrically just as good as any other.
And not only was the progress of geometry thus retarded
but that of algebra, too, in as great or greater measure;
for not only were lines, surfaces and solids things in
imagination's realm but so, too, were numbers; it is well
known that for the Greek mathematicians and a long
series of their successors, numbers were geometric things
-one number was a lne (segment), a product of two
numbers was a rectangle or a square, and that of three
a parallelopiped or a cube. False psychology thus
balked the advancement of equation theory, for x3 was
indeed a real thing,-a geometric cube,-but what, pray,
was x4, for example, or x5, and so on? The answer was
evident-they were unreal. So, in the sixteenth century,
we are told by Stifel, reviser of Rudolph's Algebra, that
"going beyond the cube just as if there were more than
three dimensions" is a thing "against nature." And in
the following century John Wallis regards the giving of
"ilngeometrical" names to the fourth and higher powers
of numbers as quite intolerable. They are, he says, a
"Monster in Nature, less possible than a Chimaera or
Centaure." Why? Because "Length, Breadth and
Thickness take up the whole of Space. Nor can Fansie
imagine how there should be a Fourth Local Dimension
beyond these Three." Here note again what the barrier
is,-a childishly naïve psychology,-Nature does not
transcend the imaginable,-what is merely conceivable is
monstrous,-a psychology so rude and so crude that, were