Engels moves back, then, to mathematics, and the points made earlier in relation to the calculus. The argument used by Engels, here, follows that set out above in relation to the barley and insects. In other words, the negation of the barley grain is effected by means of it being transformed into a plant, and, thereby, raised to a higher power, as it is multiplied into many such grains.
So, he takes an algebraic quantity a, whose negation is, thereby, not zero, but -a, i.e. (not a). To think of a line of code – X = X + 1, i.e. X = -X (not X). This not X could equally be X2, or X – 1, and so on. The minus sign, in formal logic, does not mean minus, but "not". To take the barley seed, its negation is its transformation into the barley plant, and it is out of this process that the negation of the negation results in numerous grains of barley. Similarly, the negation of a is -a, and the negation of that negation is the the multiplication of -a, -a2, raising it to a higher power, but, in maths, -a2 is a2.
“It makes no difference in this case that we can obtain the same a2 by multiplying the positive a by itself, thus likewise getting a2. For the negated negation is so securely entrenched in a2 that the latter always has two square roots, namely, a and —a. The fact that it is impossible to get rid of the negated negation, the negative root of the square, acquires very obvious significance as soon as we come to quadratic equations.” (p 174)
Turning to the calculus, “those “summations of indefinitely small magnitudes” which Herr Dühring himself declares are the highest operations of mathematics” (p 174-5), Engels describes its use.
“In a given problem, for example, I have two variables, x and y, neither of which can vary without the other also varying in a ratio determined by the facts of the case. I differentiate x and y, i.e., I take x and y as so infinitely small that in comparison with any real quantity, however small, they disappear, that nothing is left of x and y but their reciprocal relation without any, so to speak, material basis, a quantitative ratio in which there is no quantity. Therefore, dy/dx, the ratio between the differentials of x and y, is equal to 0/0 but 0/0 taken as the expression of y/x.” (p 175)
Of itself, this is an absurdity. Nothing real can have a size/quantity of zero. A piece of wood with a length of zero metres is a piece of wood that does not exist, other than as an abstraction. Yet, even for elementary mathematics, we can't make simple calculations without zero. This same zero, in the context of the calculus, is the same as the point of tangency, where a straight line is equal to a curved line. In practice, what defining the point as a line of zero length does is to overcome the problem that formal logic has with dealing with the contradiction that, at this point, A is simultaneously -A, a straight line is also a curved line. The cost of avoiding that contradiction is to posit an abstract concept, a point of zero size, that cannot exist in the real world.
“Now, what have I done but negate x and y, though not in such a way that I need not bother about them any more, which is the way metaphysics negates, but in the way that corresponds with the facts of the case? In place of x and y, therefore, I have their negation, dx and dy, in the formulas or equations before me. Now I continue to operate with these formulas, treating dx and dy as quantities which are real, though subject to certain exceptional laws, and at a certain point I negate the negation, i.e., I integrate the differential formula, and in place of dx and dy again get the real quantities x and y, and then am not where I was at the beginning, but on the contrary have in this way solved the problem on which ordinary geometry and algebra might perhaps have broken their jaws in vain.” (p 175)
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