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 f