HKB
restoration of a piece of ground made up of exceedingly diverse chemical elements to a condition of mechanical pulverisation, which is the cause of a most abundant and diverse vegetation.
It is the same also in mathematics. Let us take an ordinary algebraic quantity a. Let us negate it, then we have-a (minus a). Let us negate this negation, that is let us multiply --a by --a and we have + a^2, that is the original positive quantity but in a higher form that is to the second power. It does not matter that we can attain the same a^2 by the multiplication of a positive by itself. The negated negation is established so completely in a^2 that under all circumstances it has two square roots a and --a. And this impossibility, the negated negation, the getting rid of the negative root in the square has much significance in quadratic equations. The negation of the negation is more evident in the higher analyses, in those "unlimited summations of small quantities," which Herr Duehring himself explains as being the highest operations of mathematics and which are usually called the differential and integral calculus. How do these forms of calculation fulfil themselves? I have for example in a given problem two variable quantities x and y, of which one cannot vary without causing the other to vary also under fixed conditions. I differentiate x and y, that is I consider x and y as being so infinitesimally small that they do not represent any real quantities, even the smallest, so that, of x and y nothing remains, except their reciprocal relations, a quantitative relation without any quantity; therefore dx/dy, the relation of the two differentials of x and y, is 0/0 but 0/0 is fixed as the expression of y/x. That this relation between two vanished quantities, the fixed moment of their vanishing, is a contradiction I merely mention in passing, it should give us as little uneasiness as it has given mathematics for the two hundred or so years past. What have I done except to negate x and y; not as in metaphysics so as not to trouble myself any further about them, but in a manner demanded by the problem? Instead of x and y, I have therefore their negation dx and dy in the formulae or equations before me. I now calculate further with these formulae. I treat dx and dy as real quantities, as quantities subject to certain exceptional laws, and at a certain point I negate the negation, that is, I integrate the differential formula. I get instead of dx and dy the real quantities x and y again, and am thereby no further forward than at the beginning, but I have thereby solved the problem over which ordinary geometry and algebra would probably have gnashed their teeth in vain.
It is not otherwise in history. All civilised peoples began with common property in land. Among all peoples which pass beyond a certain primitive stage the common property in land becomes a fetter upon production in the process of agricultural development. It is cast aside, negated, and, after shorter or longer intervening periods, is transformed into private property. But at a higher stage, through the development still further of agriculture, private property becomes in its turn a bar to production, as is to-day the case with both large and small land proprietorship. The next step, to negate it in turn, to transform it into social property, necessarily follows. This advance however does not signify the restoration of the old primitive common property, but the establishment of a far higher better developed form of communal proprietorship, which, far from being an impediment to production, rather, for the first time is bound to put an end to its limitations and to give it the full benefit of modern discoveries in chemistry and mechanical inventions.