A Minor Point on Infinite Series

All too often mathematicians and theoretical physicists use infinity as a kind of mathematical rug under which they can sweep their algebraic dirt. But we need to have mathematics as a logically clean structure, so we need to remove the dirt from under the rug and dispose of it properly. As one who bears as much guilt in this matter as does anyone else, I take upon myself the task of doing what I can to remove that dirt.

Consider the usual technique of summing an infinite geometric series. We have, for the simplest example,

(Eq'n 1)

The ellipsis at the end of the series sum indicates that the terms comprise an infinite set. We sum that series by first calculating the product xS, then calculating S-xS=1, and get

(Eq'n 2)

That gives us the correct result for 0≤x≤1, but gives the wrong result for x>1. We have supposed, as a tacit axiom underlying mathematics, that mathematical logic will never give us a wrong result. In conformity with that axiom, we must assume that we have made an error in our reasoning. To explore that possibility, let's sum a finite geometric series.

We have for our new series

(Eq'n 3)

As we did before, we multiply S_{n} by x, subtract the
result from S_{n}, and divide by 1-x. This time we obtain

(Eq'n 4)

We now have an extra term in the numerator of our expression and that extra term corrects our error. If we take the limit of that equation as we let n go to infinity, then for any x≤1 we get the same result we get from Equation 2, but for any x>1 we get an infinite result, as we should, rather than the finite negative number that we calculate from Equation 2.

As we saw in our treatment of Georg Cantor's infinities, we do well to express our ideas in finite terms first and then determine the limit of our results as we allow the appropriate parameter to go to infinity. In that way we remove the possibility of using infinity as a kind of magic number with which we can create the illusion that mathematics actually fulfills our needs, rather than showing us a reality beyond our control. And here I will reiterate my Platonic view that once we have defined the numbers as an ordered sequence of names, we have made mathematics a purely objective science. We have the freedom to choose which problems to solve, but we have no freedom to choose what solutions we will obtain. Thus, if we carry out a calculation that gives inconsistent results, then we can infer that we have made an error in our use of mathematical logic. By seeking out and correcting such errors we increase the strength of the structure that we call mathematics.

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