on merely indirectly in a round about way to express the
extent of the universe. Herr Duehring on the contrary forces us to
accept his six dimensions of space and at the same time has no words
in which to express his contempt of the mathematical mysticism of
Gauss who would not content himself with the three dimensions of
space.

Applied to time, the series or row of objects, infinite at both
extremities, has a certain figurative significance. But let us picture
time as proceeding from unity or a line proceeding from a fixed point.
We can say then that time has had a beginning. We assume just what we
wanted to prove. We give a one-sided half-character to infinity of
time. But a one-sided eternity split in halves is a contradiction in
itself, the exact opposite of a hypothetical infinity, incapable of
contradiction. We can only overcome this contradiction by assuming
that the unity which we began to count the progression from, the point
from which we measure the line, is a unity taken at pleasure in the
series, a point taken at pleasure in the line. Hence as far as the
line or series is concerned it is immaterial where we put it.

But as for the contradiction of the "counted endless progression" we
shall be in a position to examine it more closely as soon as Herr
Duehring has taught us the trick of reckoning it. If he has
accomplished the feat of counting from minus infinity to zero, we
shall be glad to hear from him again. It is clear that wherever he
begins to count he leaves behind him an endless progression, and with
it the problem which he had to solve. Let him only take his own
infinite progression 1 + 2 + 3 + 4 etc. and try to reckon back to 1
again from the infinite end. He evidently does not comprehend the
requirements of the problem. And furthermore, if he affirms that the
infinite progression of past time is capable of calculation he must
affirm that time has a beginning for otherwise he could not begin to
calculate. Therefore he again substitutes a supposition for what he
had to prove. The idea of the calculated infinite series, in other
words Duehring's all-embracing law of the fixed number, is therefore a
contradiction in adjecto, is a self contradiction, and an absurd one,
moreover.

It is clear that an infinity which has an end but no beginning is
neither more nor less than an infinity which has a beginning but no
end. The least logical insight would have compelled Herr Duehring to
the statement that