HKB
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 concept that the form of a cylinder was derived from the revolution of a rectangle round one of its sides, he must have examined a number of rectangles and cylinders even if of imperfect form. Like all sciences, mathematics has sprung from the necessities of men, from the measurement of land and the content of vessels, from the calculation of time and mechanics. But, as in every department of thought, at a certain stage of development, laws are abstracted from the actual phenomena, are separated from them and set over against them, as something independent of them, as laws, which apparently come from the outside, in accordance with which the material world must necessarily conduct itself. So, it has happened in society and the state, so, and not otherwise, pure mathematics though borrowed from the world is applied to the world, and though it only shows a portion of its component factors is all the better applicable on that account.