 What is often pointed out in response is that Zeno gives us no reason to think that the sum is infinite rather than finite. He might have had the intuition that any infinite sum of finite quantities, since it grows endlessly with each new term must be infinite, but one might also take this kind of example as showing that some infinite sums are after all finite. Thus, contrary to what he thought, Zeno has not proven that the absurd conclusion follows. However, what is not always appreciated is that the pluralist is not off the hook so easily, for it is not enough just to say that the sum might be finite, she must also show that it is finite—otherwise we remain uncertain about the tenability of her position. As an illustration of the difficulty faced here consider the following: many commentators speak as if it is simply obvious that the infinite sum of the fractions is 1, that there is nothing to infinite summation. But what about the following sum: \(1 - 1 + 1 - 1 + 1 -\ldots\). Obviously, it seems, the sum can be rewritten \((1 - 1) + (1 - 1) + \ldots = 0 + 0 + \ldots = 0\). Surely this answer seems as intuitive as the sum of fractions. But this sum can also be rewritten \(1 - (1 - 1 + 1 - 1 +\ldots) = 1 - 0\)—since we’ve just shown that the term in parentheses vanishes—\(= 1\). Relying on intuitions about how to perform infinite sums leads to the conclusion that \(1 = 0\). Until one can give a theory of infinite sums that can give a satisfactory answer to any problem, one cannot say that Zeno’s infinite sum is obviously finite. Such a theory was not fully worked out until the Nineteenth century by Cauchy. (In Cauchy’s system \(1/2 + 1/4 + \ldots = 1\) but \(1 - 1 + 1 -\ldots\) is undefined.)