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A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. However, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.
Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno *started* with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space.
Put a different way, when these kinds of solutions tell us that Achilles passes the tortoise 10/9 meter after the tortoise's starting point, they assume that Achilles can actually reach that point, but Zeno questioned that Achilles can actually ever get to that point. Similarly, when we are told that Achilles passes the tortoise 10/9 seconds into the race, it is assumed that time can actually flow to that point, but once again we get the same problem: If there are an infinite number of time points between t = 0 and t = 10/9, how can t = 10/9 ever be reached? How, indeed, can time flow at all if it is assumed that between any two time points there are infinitely many other time points that, at least under our naive conception of time, have to occur one after the other? Thus, all these kinds of solutions presuppose that Zeno's difficulties have already been solved when trying to resolve the paradox. Which is to say, they beg the question and therefore don't resolve anything at all.
An unfortunate complication among these kinds of discussions is that many treatments of Zeno's paradox present Zeno's reasoning in such a way that calculus and series-based solutions really do work as objections to that reasoning. To be precise, Zeno's reasoning is often presented as arguing that because there are an infinite number of tasks to be done, it will take an infinite amount of time to complete all these tasks, and the calculus and mathematical series based solutions are now perfectly correct in objecting to that argument by pointing out that the sum of an infinite number of time intervals can add up to a finite amount of time. However, such a presentation of Zeno's argument makes the argument into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler argument that does not at all consider any quantifications of time. This much simpler argument simply states that for Achilles to capture the tortoise an infinite series of physical events need to be completed, which is logically impossible. The calculus and mathematical series based solutions offer no insight into this much simpler, much more stinging, paradox.[2]
The following thought experiment can be used to illustrate the fact that time is irrelevant to the paradox. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle at all, then apparently Achilles must have stopped counting, and we could ask Achilles what the greatest number he counted to was. But of course this is nonsense: there is no greatest number, and Achilles can never stop counting. So, Achilles can't catch up with the Tortoise, whether he has finite time or infinite time to do so.
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