ole of pure mathematics without making any use of the experience
which is afforded us in the objective world. In pure mathematics the
understanding is engaged "in its own free creations and imaginations";
the concepts of number and form are "self-sufficient objects
proceeding from themselves" and so have "a value independent of
individual experience and actual objective reality."

That pure mathematics has a significance independent of particular
individual experience is quite true as are also the established facts
of all the sciences and indeed of all facts. The magnetic poles, the
formation of water from oxygen and hydrogen, the fact that Hegel is
dead and that Herr Duehring is alive, are facts independent of my
experience or that of any other single individual, and will be
independent of that of Herr Duehring himself, as soon as he shall
sleep the sleep of the just. But in pure mathematics the mind is not
by any means engaged with its own creations and imaginings. The
concepts of number and form have only come to us by the way of the
real world. The ten fingers on which men count and thereby performed
the first arithmetical calculations are anything but a free creation
of the mind. To count not only requires objects capable of being
counted but the ability, when these objects are regarded, of
subtracting all qualities from them except number and this ability is
the product of long historical development of actual experience. The
concept form is, like that of number, derived exclusively from the
external world and is not a purely mental product. To it things
possessed of shape were necessary and these shapes men compared until
the concept form was arrived at. Pure mathematics considers the shapes
and quantities of things in the actual world, very real objects. The
fact that these objects appear in a very abstract form only
superficially conceals their origin in the world of external nature.
In order to understand these forms and qualities in their purity it is
necessary to separate them from their content and thus one gets the
point, without dimensions, the line, without breadth and thickness, a
and b, x and y, constants and variables, and we finally first arrive
at independent creations of the imagination and intellect, imaginary
magnitudes. Also the apparent derivation of mathematical magnitudes
from each other does not prove their aprioristic origin, but only
their rational interconnection. Before one attained the