This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough:
A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to the wall. He, however, is to walk half the distance first, then half the remaining distance, then half of that, then half of that and so on. This means the man will never get to the wall, as there's always a half remaining.
This defies common sense. We know a man(or woman, of course) can just walk up to a wall and get to it.
My calculus book says this was solved when the idea of a limit of an infinite series was developed. The idea says that if the distances the man is passing are getting closer and closer to the total area from where he started to the wall, this means that the total distance is the limit of that.
What I don't understand is this: mathematics tells us that the sum of the infinite little distances is finite, but, in real life, doesn't walking an infinite number of distances require an infinite amount of time, which means we didn't really solve anything, since that's what troubled Zeno?
Thanks.
Best Answer
It is often claimed that Zeno's paradoxes of motion were "resolved by" the infinitesimal calculus, but I don't really think this claim stands up to a closer investigations. It depends on very specific notion of what it would mean to "resolve" the paradox, and it looks to me that this notion can only be arrived at if one deliberately sets out to reformulate the paradox into a problem that
In this way the original "therefore Achilles never overtakes the tortoise!" becomes tacitly transformed into "WHEN does Achilles overtake the tortiose?" and once we get a numerical answer to the latter, the popularizers will claim that the original contradiction has been resolved.
Actually, the first step in resolving the paradox must be to understand why it was supposed to be paradoxical in the first place. Zeno's paradoxes rely on an intuitive conviction that
Only if we accept this claim as true does a paradox arise. But why should we accept that as true? If we reject it, then the paradox disappears -- and we don't need any infinitesimal calculus, or any concept of limit, to deny this premise. On the contrary, it must be up to anyone who wants us to accept it to provide some kind of argument -- and then he has the trouble of explaining why his pet principle doesn't conflict with reality, as demonstrated by Zeno!
Part of the story here is that the historical record is too fragmentary for us to know what the historical Zeno actually intended with the paradoxes. None of his own writings survive, and all we have to go by is other classical authors, who discussed the paradoxes for their own purposes, and just gave credit to Zeno for coming up with the example.
On this sparse record, it is entirely possible that Zeno never intended to convince anyone that motion is impossible. Instead, it is easy to imagine that, for example, one of Zeno's pupils may have tried to use the indented principle above in an argument, and then the master immediately shot that down by using the "paradox" as a vivid practical example of how there can be infinitely many instants or intervals in a finite period of time.