HKB
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 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.