But even in the latter there is a host of contradictions. It is a contradiction, for example, that a root of A should be and actually is a power of A. A to the power of one-half equals the square root of A. It is contradiction that a negative magnitude should be the square of anything, since every negative magnitude multiplied by itself gives a positive square. The square root of minus one is therefore not only a contradiction but an absurd contradiction, a veritable absurdity. And yet the square root of minus one is in many instances the necessary result of correct mathematical operations, nay further, where would mathematics higher or lower be if one were forbidden to operate with the square root of minus one.
Mathematics itself enters the realm of the dialectic and significantly enough it was a dialectic philosopher, Descartes, who introduced this progressiveness into mathematics. As is the relation of the mathematics of variable magnitudes to that of invariable quantities, so is the relation of the dialectic method of thought to the metaphysical. This does not prevent the great majority of mathematicians from only recognising the dialectic in the realms of mathematics, a condition of things satisfactory to those who operate in the antiquated, limited, metaphysical fashion by methods attained by means of the dialectic.
(Duehring having made an attack upon Marx's "Capital" because of its reliance upon the dialectic, and having indulged in the epithets to which he is too prone with respect to this work, Engels takes up its defence in that respect as follows):
It is not our business to concern ourselves at this point with the correctness or incorrectness of the investigations of Marx as regards economics, but only with the application which he makes of the dialectic method. So much is certain, that it is only now that the readers of "Capital" will by the aid of Herr Duehring understand what they have read properly, and among them Herr Duehring himself, who in the year 1867 was still in a position, as far as possible to a man of his calibre, to review the book rationally. He did not then, it may be noted, first translate the arguments of Marx into Duehringese, as now seems indispensable to him. Even if he at that time made the blunder of identifying the Marxian dialectic with that of Hegel he had not altogether lost the ability to distinguish methods from the results attained by them and to comprehend that an abuse of the former is no contradiction of the latter.
Herr Duehring's most astonishing observation is that from the Marxian standpoint, "in the last analysis everything is identical," that therefore in the eyes of Marx, for example, capitalists and wage workers, feudal, capitalistic and social methods of production are "all one." In order to show the possibility of such sheer stupidity it only remains to point out that the mere word "dialectic" makes Herr Duehring mentally irresponsible and makes what he says and does so inaccurate and confused as to be in the last analysis "all one."
(Herr Duehring remarks, "How comical for example is the declaration based upon Hegel's confused notions that quantity becomes lost in quality and that money advanced [i.e. for productive purposes. Ed.] becomes capital when it reaches a certain limit merely through quantitative increase." To which Engels replies thus):
This seems peculiar when presented in this washed out fashion by Herr Duehring. On page 313 (2nd ed. "Capital") Marx, after an investigation of fixed and variable capital and surplus value, derives from his investigations the conclusion that "not every amount of gold or value capable of being transformed into capital is so transformed; rather a certain minimum of gold or of exchange value is presupposed to be in the possession of the individual owner of gold or goods." He thereupon gives an example, thus, in a branch of industry the worker works eight hours per day for himself, i.e. in order to produce the value of his wages, and the following four hours for the capitalist in producing surplus value to go into their pockets. One must have sufficient values to permit of the setting up of two workmen with raw material, means of labor and wages, in order to live as well as a workman. But since capitalistic production is not undertaken for mere livelihood but for increase of wealth, our individual with his two workmen would still be no capitalist. If he lives twice as well as an ordinary workman and transforms half of the surplus value produced into capital he will have to employ eight workmen and possess four times the aforementioned amount of value, and only after this and other examples for the purpose of illustrating and establishing the fact that not every small amount of value can effect a transformation of itself into capital, but that each period of industrial development and each branch of industry has its own minimum, fixed, Marx remarks "Here, as in nature, the correctness of the law of logic, as discovered by Hegel, is established—that mere quantitative changes at a certain point suddenly take on qualitative differences."
One may remark the elevated and dignified fashion in which Duehring makes Marx say the exact opposite of what he did say. Marx says "The fact that a given amount of value can only transform itself into capital as soon as it has attained a definite minimum, varying with circumstances, in each individual case,—this fact is proof of the correctness of the law of Hegel. Herr Duehring makes him say "Because, according to the law of Hegel, quantity is transformed into quality therefore 'a sum of money when it has reached a certain amount becomes capital.'" He says just the opposite.