HKB
in the persons of human beings with limited faculties and powers of knowledge, is dissolved in the, for us at least, practically endless progression of the race, in unending progress.
We stated just now that higher mathematics holds as one of its basic principles that straight and crooked may be identical under certain circumstances. It shows another contradiction, that lines which apparently intersect yet are parallel from five to six centimeters from the point of intersection, should be such as should never intersect although indefinitely produced, and yet, notwithstanding these and even greater contradictions, it produces not only correct results but results which are unattainable by lower mathematics.
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."