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:
