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NOTES ON MARXISM
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N M MN DeKoninck-07 5/13/09 3:54 PM Page 379 DeKoninck-07 5/13/09 3:54 PM Page 380 [34.205.246.61] Project MUSE (2024-03-29 00:00 GMT) I. Calculus and Dialectic In the Anti-Dühring as well as in The Dialectics of Nature, Engels recognized the fundamentally dialectical character of the calculus.1 “[T]he mathematics of variable magnitudes, of which the most important part is the infinitesimal calculus, is essentially nothing other than the application of dialectics to mathematical questions.”2 “Moreover, I don’t think that one can give the calculus as an example of the sort of contradiction that dialectic taken in the Hegelian or Marxist sense demands, that is, a contradiction objectively existing in things and phenomena themselves. . . .”3 J.B.S. Haldane alludes to this.4 Let’s examine the question in more detail in order to assure ourselves of the direction and degree of the concession that the Marxist seems compelled to make today. The problem is not without interest since Engels saw in it,as well as in the case of infinity and movement, a glaring example of a contradiction in the Marxist sense. But if the Marxist on the one hand calls calculus dialectical because of this contradiction, and if on the other hand we must now recognize that we do not have there the sort of contradiction that Marxist dialectic demands—at least not at the level indicated by Engels—how does the Marxist understand the dialectical character of the calculus? Should he say that the calculus is no longer dialectical and that this claim must now be consigned to history? To come to Engels’aid by saying that in this manner he was only basing himself on the state of mathematics in his time would be,I am persuaded, unjust to him. Here is a relevant passage that I take from the Anti-Dühring: I distinguish x and y: that is, I suppose that x and y are infinitely small and that they disappear in relation to any magnitude no matter how small, however little one posits it as really existing, in such wise that of x and y there exists only their reciprocal relation without, so to say, any material foundation, a quantitative relation without any quantity. The expression dy/dx, that is, the relation of two differentials of x and of y is therefore equal to /, but this / is posited as the expression of y/x. I note in passing that this relation between two disappeared magnitudes, DeKoninck-07 5/13/09 3:54 PM Page 381 fixing the moment of their disappearance, implies a contradiction; but this contradiction should not trouble us any more than it troubled mathematicians for more than two hundred years.5 This interpretation calls into question the notion of limit. But, in terms of the method of limits, it could be a question of a contradiction in the example given by Engels only if he supposes that the limit is truly attained. But that is contrary to the very definition of limit: the successive values of the variable x must approach a fixed given number in such a way that the difference x – a ends by becoming and remaining a value less than any given number , e, no matter how small. It is essential to the definition of the derivative that dx, that is x – a, differs from . In order for a contradiction to appear in the example given, the magnitudes in question would have to be determinately equal to ; that they not only be magnitudes on the way to disappearing and becoming equal to , but that they have entirely disappeared and are actually equal to . We know that the first interpreters of the calculus had a false understanding of the infinitely small and returned, in effect, to the ancient error of Antiphon. One need only consider the infinitely small as a static entity for it to become as contradictory as a square circle. In fact, it is contradictory only if one takes away its dynamic character.As infinitely small,it has no definite magnitude. No doubt it is something negative with respect to a definite magnitude from which it departs.As Engels himself would say: it disappears with reference to any magnitude.But it remains in the state of “disappearing,” and what has disappeared always refers to a given magnitude from which it departs. Since it is never equal to , the infinitely...