Friedrich Engels - Landmarks of Scientific Socialism - Anti-Duehring

Further, if no philosophy, as such, is longer required there is no longer the necessity of any philosophy of nature even. The view that all the phenomena of nature stand in systematic mutual relations compels science to prove this systematic interconnection in all respects, in single cases as well as in the entirety. But an appropriate creative, scientific representation of this mutual connection in such a way as to show the composition of an exact thought-picture of the system of the universe in which we live remains not only for us but for all time an impossibility. Should such a final conclusive system of the interconnection of the various activities of the universe, physical, as well as intellectual and historical, ever be brought to completion at any point of time in the history of the human race, human knowledge would forthwith come to an end and future historical progress would be cut off from the very moment in which society was directed in accordance with the system, which would be an absurdity, mere nonsense.

Man is therefore confronted by a contradiction, on the one hand he is obliged to study the interconnections of the world-system exhaustively, and, on the other hand, he is unable to fully accomplish the task either as regards himself or as regards the system of nature. This contradiction, however, does not consist solely in the nature of the two factors World and Man; it is the main lever also of universal intellectual progress and is solved every day and for ever in an endless progressive development of humanity, just as mathematical problems find their solution in an endless progression of a recurring decimal. As a matter of fact also every concept of the universe is subject to objective limitations owing to the conditions of historical knowledge, and subjectively in addition owing to the physical and mental make up of the author of the concept. But Herr Duehring exhibits a mode of thought which is confined in its application to a limited and subjective idea of the universe. We saw earlier that he was omnipresent, in all possible forms of the universe, now we see that he is omniscient. He has solved the final problems of science and has nailed up tight all future knowledge.

Herr Duehring considers that he can, as with the fundamental forms of existence, produce aprioristically by means of his own cogitations the whole 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 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 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 shown, a conclusion, upon the correctness of which all logic depends, and which is demonstrated therefore outside of pure mathematics. The remaining axioms with regard to equality and inequality are merely logical extensions of this conclusion. Such barren statements are not enticing either in mathematics or anywhere else. To proceed we must have realities, conditions and forms taken from real material things; representations of lines, planes, angles, polygons, spheres, etc., are all borrowed from reality, and it is just naive ideology to believe the mathematicians, who assert that the first line was made by causing a point to progress through space, the first plane by means of the movement of a line, and the first solid by revolving a plane, etc. Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid – corpus solidum – and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.