The problem itself has a very ****** solution. Eternity in time, infinity in space, signify from the start, and in the ****** meaning of the words, that there is no end in any direction neither forwards nor backwards, upwards or downwards, to the right or to the left. This infinity is something quite different from that of an infinite series, for the latter always starts from one, with a first term. The inapplicability of this idea of series to our object becomes clear directly we apply it to space. The infinite series, transferred to the sphere of space, is a line drawn from a definite point in a definite direction to infinity. Is the infinity of space expressed in this even in the remotest way? On the contrary, the idea of spatial dimensions involves six lines drawn from this one point in three opposite directions, and consequently we would have six of these dimensions. Kant saw this so clearly that he transferred his numerical series only indirectly, in a roundabout way, to the space relations of the world. Herr Dühring, on the other hand, compels us to accept six dimensions in space, and immediately afterwards can find no words to express his indignation at the mathematical mysticism of Gauss, who would not rest content with the usual three dimensions of space [37] {See D. Ph. 67-68}.
As applied to time, the line or series of units infinite in both directions has a certain figurative meaning. But if we think of time as a series counted from one forward, or as a line starting from a definite point , we imply in advance that time has a beginning: we put forward as a premise precisely what we are to prove. We give the infinity of time a one-sided, halved character; but a one-sided, halved infinity is also a contradiction in itself, the exact opposite of an "infinity conceived without contradiction". We can only get past this contradiction if we assume that the one from which we begin to count the series, the point from which we proceed to measure the line is any one in the series, that it is any one of the points in the line, and that it is a matter of indifference to the line or to the series where we place this one or this point.
But what of the contradiction of "the counted infinite numerical series"? We shall be in a position to examine this more closely as soon as Herr Dühring has performed for us the clever trick of counting it. When he has completed the task of counting from - = (minus infinity)to 0 let him come again. It is certainly obvious that, at whatever point he begins to count, he will leave behind him an infinite series and, with it, the task which he is to fulfil. Let him just reverse his own infinite series 1 + 2 + 3 + 4 ... and try to count from the infinite end back to 1; it would obviously only be attempted by a man who has not the faintest understanding of what the problem is. And again: if Herr Dühring states that the infinite series of elapsed time has been counted, he is thereby stating that time has a beginning; for otherwise he would not have been able to start "counting" at all. Once again, therefore, he puts into the argument, as a premise, the thing that he has to prove.
The idea of an infinite series which has been counted, in other words, the world-encompassing Dühringian law of definite number, is therefore a contradictio in adjecto ["contradiction in definition"-- ed.] contains within itself a contradiction, and in fact an absurd contradiction.
It is clear that an infinity which has an end but no beginning is neither more nor less infinite than that which has a beginning but no end. The slightest dialectical insight should have told Herr Dühring that beginning and end necessarily belong together, like the north pole and the south pole, and that if the end is left out, the beginning just becomes the end -- the one end which the series has; and vice versa.
The whole deception would be impossible but for the mathematical usage of working with infinite series. Because in mathematics it is necessary to start from definite, finite terms in order to reach the indefinite, the infinite, all mathematical series, positive or negative, must start from 1, or they cannot be used for calculation. The abstract requirement of a mathematician is, however, far from being a compulsory law for the world of reality.
For that matter, Herr Dühring will never succeed in conceiving real infinity without contradiction. Infinity is a contradiction, and is full of contradictions. From the outset it is a contradiction that an infinity is composed of nothing but finites, and yet this is the case. The limitedness of the material world leads no less to contradictions than its unlimitedness, and every attempt to get over these contradictions leads, as we have seen, to new and worse contradictions. It is just because infinity is a contradiction that it is an infinite process, unrolling endlessly in time and in space. The removal of the contradiction would be the end of infinity. Hegel saw this quite correctly, and for that reason treated with well-merited contempt the gentlemen who subtilised over this contradiction.
