Hegel and Infinite Series
Posted by Tanas Gjorgoski on June 28, 2006
Connected to the last post about comprehending how 1=0.99(9), here is what Hegel (§ 561 , The Science of Logic read at own risk) has to say about the issue of infinite series (in his own specific way):
Thus the usually so-called sum, the 2/7 or 1/(1 – a) is in fact a ratio; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate.
Furthermore, the series is in fact the finite expression; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be; and then, too, what it lacks is itself a specific quantum; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted; this is a kind of superstition, the superstition of the understanding; we have seen how, on the contrary, it indicates only a deficiency.
Let me try to explain previous quote with an example:
We say that 1+a+a2+…=1/(1-a) for a<1. But how to understand that addition of infinite number of terms? We can add as much terms as we like, but the sum will never be equal to the ratio. What does this equation say then?
If we subtract 1 from the ratio (which is supposed to be sum of the infinite series) we have:
1/(1-a) – 1 = (1-1+a)/(1-a) = a·[1/(1-a)] or by moving the 1 to the right side…
1/(1-a) = 1 + a·[1/(1-a)]…
So, now we got to a recursive formula of the form
A = 1 + a·A , where A is 1/(1-a)
And by iterating we get:
A = 1 + a·A
, then by changing second A with the whole right side we get…
A = 1 + a·(1 + a·A) = 1 + a + a2·A
, we change again, and we get…
A = 1 + a + a2·(1 + a·A) = 1 + a + a2 + a3·A
and so on, and so on ad infinitum.
We see that the sequence in the series can be generated from the ratio by recursive process, but it can never end, we can generate as much terms as we like, there will always be a certain left over (an·A), which also needs to be expressed through new terms.
So Hegel is saying that while we can represent what is specified by the ratio through infinite series, the series will be a “deficient” representation of what is clearly contained in the ratio itself.