Page:Landmarks of Scientific Socialism-Anti-Duehring-Engels-Lewis-1907.djvu/63

 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 of content all the more strongly.

(2) If two magnitudes are equal to a third they are equal to one another; this statement is, as Hegel has