beginning and end are mutually necessary to each
other, like North Pole and South Pole, and that if one omit the end
the beginning becomes the end, the one end which the series has and
vice versa.

The entire fallacy would not be possible if it were not for the
mathematical practice of operating with an infinite series. Because in
mathematics one must proceed from the given and finite to that which
is not given and infinite, all mathematical series whether positive or
negative, begin with a fixed point otherwise one cannot calculate. The
ideal necessities of the mathematician however are very far from being
a law compulsory upon the universe.

Besides Herr Duehring will never succeed in imagining an infinity
without contradiction. In the first place, infinity is a contradiction
and full of contradictions. For example it is a contradiction that
infinity should be made up of finite things and yet such is the case.
The notion of a limited universe leads to contradictions just as much
as the notion of its unlimitedness, and each attempt to abolish these
contradictions leads, as we have seen, to new and worse
contradictions. But just because infinity is a contradiction, it is
without end, endlessly developing itself in time and space. The
abolition of the contradiction would be the end of infinity. Hegel
saw that very clearly, and covers the people who entered upon
intricate arguments about this contradiction with merited scorn.

Let us proceed. Now, time has had a beginning. What was before this
beginning? The unchangeable universe incomparable with anything else.
And as no changes occur in this condition the particular concept time
is transformed into the general concept existence. In the first place
we have nothing to do with the transformation which goes on in the
brain of Herr Duehring. We are not engaged with a concept of time, but
with actual time of which Herr Duehring cannot so easily dispose. In
the second place no matter how much the concept of time is transformed
into the general concept existence it does not bring us one step
nearer the goal. For the fundamental forms of all existence are space
and time, and a thing existing outside of time is as silly an idea as
that of a being outside of space. The Hegelian "past existence in
which there was no time" and the neo-Schelling "being beyond the scope
of thought" are rational conceptions compared with this being outside
of time. For this reason Herr Duehring goes