Engels notes that there is contradiction in the fact that the definition of parallel lines never meet or cross, and, yet, in the real world we see parallel lines that do, indeed, cross. In fact, the Mobius Strip, which also provides the solution to the three utility puzzle, also has practical application, in relation, for example, to the wear and tear of conveyor belts, ensuring even wear on both sides of the belt.
Even in relation to lower mathematics, it relies on contradiction, as, for example, with the use of imaginary numbers based on the root of -1.
“And yet √-1 is in many cases a necessary result of correct mathematical operations. Furthermore, where would mathematics — lower or higher — be, if it were prohibited from operation with √-1 ?” (p 154)
In the realm of computer programming, the use of variable numbers is fundamental. Just to undertake a simple counting operation, the line of code reads X = X +1. That is, itself, an expression of expansion, and movement, of X being simultaneously itself and not itself. The use of variable magnitudes in maths was introduced by Descartes, himself a dialectical philosopher.
“The relation between the mathematics of variable and the mathematics of constant magnitudes is in general the same as the relation between dialectical and metaphysical thought. Which by no means prevents the great mass of mathematicians from recognising dialectics solely in the sphere of mathematics, and a good many of them from continuing to work in the old, limited, metaphysical way with methods that were obtained dialectically.” (p 154)
In contrast to this unacknowledged use of results obtained by dialectical methods, Duhring, whilst vocally condemning dialectics, provides no actual examples of results obtained from the use of his own alternative based on “antagonism of forces and his antagonistic world schematism.” (p 154)
“When Hegel’s “Doctrine of Essence” has in fact been reduced to the platitude of forces moving in opposite directions but not in contradictions, surely the best thing to do is to avoid any application of this commonplace.” (p 155)
Having attacked dialectics, this leads Duhring into his attack on Marx's theory, and its application in Capital. As with most such attacks on Marx, it is based on misrepresentation, either from ignorance or malice. Engels quotes Duhring's statement,
“The absence of natural and intelligible logic which characterizes these dialectical frills and mazes and arabesques of ideas... even to the part that has already appeared we must apply the principle that in a certain respect and also in general (!), according to a well-known philosophical prejudice, all must be sought in each and each in all, and that therefore, according to this hybrid and hobbled idea, everything is all the same in the end.” (p 155)
The attack on Capital, referring to the publication of Volume I, and an implication that the promised further volumes did not exist, and the project had been abandoned, was also a common theme of Marx's critics. Engels focuses on the last part of Duhring's statement that “everything is all the same in the end.”
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