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.

But as Herr Duehring imagines that the whole of pure mathematics can
be derived from the mathematical axioms, "which according to purely
logical concepts are neither capable of proof nor in need of any, and
without empirical ingredients anywhere and that these can be applied
to the universe, he likewise imagines, in the first place, the
foundation forms of being, the single ingredients of all knowledge,
the axioms of philosophy, to be produced by the intellect of man; he
imagines also that he can derive the whole of philosophy or plan of
the universe from these, and that his sublime genius can compel us to
accept this, his conception of nature and humanity." Unfortunately
nature and humanity are not constituted like the Prussians of the
Manteuffel regime of 1850.

The axioms of mathematics are expressions of the most elementary ideas
which mathematics must borrow from logic. They may be reduced to two.

(1) The whole is greater than its part; this statement is mere
tautology, since the quantitatively limited concept, "part,"
necessarily refers to the concept, "whole,"--in that "part" signifies
no more than that the quantitative "whole" is made up of quantitative
"parts." Since the so-called axiom merely asserts this much we are not
a step further. This can be shown to be a tautology if we say "The
whole is that which consists of several parts--a part is that several
of which make up a whole, therefore the part is less than the whole."
Where the barrenness of the repetition shows the lack