e same with mathematics. Elementary mathematics, the mathematics
of constant quantities, proceeds within the limits of formal logic, at
least as a rule: the mathematics of variable quantities which is
peculiarly concerned with calculations running to the infinite, is
substantially nothing but the application of the dialectic in
mathematics. Mere proof becomes secondary before the manifold
application of the method to new fields of investigation. But nearly
all the proofs of higher mathematics from the first of the
differential calculus, are, strictly speaking, false from the
standpoint of elementary mathematics. This cannot be otherwise, if
one, as is here the case, wishes to establish results won in the realm
of dialectics by means of formal logic. For a crass metaphysician like
Herr Duehring to want to prove anything by means of the dialectic
would be the same wasted labor as Leibnitz and his pupils went through
when they tried to establish the thesis of calculation to infinity by
means of the mathematics of their time. The differential gave them
the same spasms as the negation of the negation gives Herr Duehring
and it played a role in it as we shall see. They admitted it at last,
at least as many as did not die first, not because they were convinced
but because it always worked out right. Herr Duehring, is, as he says,
just in his forties, and if he attains old age, as we hope he will, he
may also experience the same.

But what is this dreadful negation of the negation which makes life so
bitter to Herr Duehring and which is to him what the unpardonable sin,
the sin against the Holy Ghost, is to Christianity? It is a very
simple process, and one, moreover, which fulfils itself every day,
which any child can understand when it is deprived of mystery, under
which the old idealistic philosophy found a refuge, and beneath which
it will pay unprotected metaphysicians to take refuge from the stroke
of Herr Duehring. Let us take a grain of barley. Millions of such
grains of barley will be ground, cooked and brewed and then consumed.
But let such a grain of barley fall on suitable soil under normal
conditions; a complete individual change at once takes place in it
under the influence of heat and moisture, it germinates. The grain, as
such disappears, is negated, in its place arises the plant, the
negation of the grain. But what is the normal course of life of this
plant? It grows, blossoms, bears fruit and finally produces