As described earlier, the concept of space-time, and its beginning with the Big Bang, seems to disprove this, but only if you conclude that there was nothing prior to it, or after the heat death of the universe. Some theories propose that the Big Bang was only one such Big Bang, itself one of an infinite number of such events, perhaps arising like a series of bubbles with each one being spawned from another.
“The infinite series, transferred to the sphere of space, is the 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, it requires at least six lines drawn from this one point in three opposite directions, to conceive the dimensions of space; 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 spaciality 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.” (p 61-2)
The same argument applies to time, necessarily, when we incorporate the concept of space-time. A series in relation to time, requires starting from one, but then, implies that time itself had a beginning. Its true that our current space-time may have a beginning and end, but that does not mean that time itself, as an abstraction from it has a beginning or end.
“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, 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 them.” (p 62)
Engels, then turns to Duhring's treatment of the “counted infinite numerical sequence”. Engels notes that Duhring's argument rests upon the notion that its possible to count back from any positive number to zero. Engels says,
“When he has completed the task of counting from - ∞ (minus infinity) to 0 let him come again. It is certainly obvious that, wherever he begins to count, he will leave behind him an infinite series and, with it, the task which he is to fulfil. Just let him invert 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. Still more. When Herr Dühring asserts that the infinite series of lapsed time has been counted, he is thereby asserting that time has a beginning; for otherwise he would not have been able to start “counting” at all. Once again, therefore, he smuggles into the argument, as a premise, the thing that he has to prove.” (p 62-3)
Duhring's concept of a Law of Determinate Number, therefore, contains, within itself, a contradiction in terms, a contradiction that is, itself, absurd.
“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 determinate, finite terms in order to reach the indeterminate, the infinite, all mathematical series, positive or negative, must start with 1, or they cannot be used for calculation. But the logical need of the mathematician is far from being a compulsory law for the real world.” (p 63)
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