A number of paradoxes have been attributed to Zeno. One of them is the Paradox of the Runner: in order for a runner to get to the finish line, she needs to cross the first half of the track. Once she's done that, she needs to cross half the distance from the halfway mark to the finish line. Once she's done that, she needs to cross half the distance from that point to the finish line; etc. It seems that there are infinitely many finite intervals that she needs to traverse before she makes it to the finish line. But it's impossible to accomplish in a finite amount of time infinitely many tasks, each of which takes a finite amount of time. Therefore, the racer cannot make it to the finish line.

It's common to hear that the solution is to appreciate that the sum of infinitely many finite quantities can be finite. Mathematicians have taught us, we're told, that the infinite sum:

1/2 + 1/4 + 1/8 + 1/16 + ...

actually sums to 1. So, if we view the racer as traversing the first half of the racetrack in half a minute, the next quarter of it in 15 seconds, etc., then we can see that she'll reach the finish line in exactly one minute. This result actually requires the subtleties of the calculus, a branch of mathematics that was placed on a firm footing only in the 19th Century. So it's no shame on Zeno if he didn't appreciate this solution.

Is that the end of the matter? Perhaps it is unless there remain disputes about the mathematical result. How could there be such disputes in mathematics!? Does anyone actually think that this infinite sum doesn't sum to 1? No. It's rather that not all mathematicians and philosophers would agree on how to understand the claim that this infinite sum sums to 1. When the claim is spelled out, it involves quantification over infinite totalities. And there has been substantial and difficult disagreement about how precisely to understand such quantification. For a bit more discussion, see Question 139.

A number of paradoxes have been attributed to Zeno. One of them is the

Paradox of the Runner: in order for a runner to get to the finish line, she needs to cross the first half of the track. Once she's done that, she needs to cross half the distance from the halfway mark to the finish line. Once she's done that, she needs to cross half the distance fromthat pointto the finish line; etc. It seems that there are infinitely many finite intervals that she needs to traverse before she makes it to the finish line. But it's impossible to accomplish in a finite amount of time infinitely many tasks, each of which takes a finite amount of time. Therefore, the racer cannot make it to the finish line.It's common to hear that the solution is to appreciate that the sum of infinitely many finite quantities

canbe finite. Mathematicians have taught us, we're told, that the infinite sum:1/2 + 1/4 + 1/8 + 1/16 + ...

actually sums to 1. So, if we view the racer as traversing the first half of the racetrack in half a minute, the next quarter of it in 15 seconds, etc., then we can see that she'll reach the finish line in exactly one minute. This result actually requires the subtleties of the calculus, a branch of mathematics that was placed on a firm footing only in the 19th Century. So it's no shame on Zeno if he didn't appreciate this solution.

Is that the end of the matter? Perhaps it is

unlessthere remain disputes about the mathematical result. How could there be such disputes in mathematics!? Does anyone actually think that this infinite sumdoesn'tsum to 1? No. It's rather that not all mathematicians and philosophers would agree onhow to understandthe claim that this infinite sum sums to 1. When the claim is spelled out, it involves quantification over infinite totalities. And there has been substantial and difficult disagreement about how precisely to understand such quantification. For a bit more discussion, see Question 139.