>Zeno's paradoxes are soluble by basic calculus. Once you distinguish between countable and uncountable infinities, the problem of crossing a bounded interval in finite time ceases to be paradoxical.
They're not mathematically paradoxical, but that doesn't necessarily mean that the paradoxes are solved, because there's more than just math going on. A lot of the paradoxes hinge on the question of whether it is in fact possible to traverse an infinite series of positions in space or moments in time. I have no idea whether it is or it isn't, but the issue isn't settled by calculus. Calculus allows you to figure out what the result would be if such a traversal were to occur.
No, calculus does in fact resolve them. More specifically, formalizing continuity and completeness obviates the issue. Like I said, if you distinguish between countable and uncountable infinities, there is no longer a paradox.
The only reason it appears to be paradoxical is because you're mandating someone move from a real coordinate (a, b, c) to another real coordinate (a', b', c') on the interval [x, y] while also passing through the set of all real points between them, without first defining a notion of distance of time. That's not possible for the same reason you can't ask someone to count all reals on an interval, because continuity implies uncountability. Between any pair of real numbers is another real number, and it takes an equal amount of effort (and time) to count any given number.
To a first glance, this seems like a paradox because we can clearly move from (a, b, c) to (a', b', c), yet we shouldn't be capable of any movement whatsoever. Calculus solves this problem by formalizing Zeno's demand as a geometric series with a notion of distance. The requirement is that you move from one position to another position while passing through every halfway position between them. Equip the vector space ℝ^3 with the Euclidean metric so you have a metric space (defined distance). Then we have the sequence of steps
(a, b, c) -> |(a, b, c) - (a', b', c')|/2 -> ... -> (a', b', c')
More concretely: an infinite expansion such as 0.99999999... is equal to 1. Each half step will take only half as long to traverse as the half step preceding it to it once you've defined Euclidean distance on a continuous space. The first step to formalizing sequences and series like this is by constructing the real numbers as a continuous set and distinguishing between different types of infinities. Then you can define limits, and from there you're essentially done.
Note that at no point am I talking about what happens when you reach 1, or (a', b', c'), or anywhere else. I'm just explaining how you reach it in finite time. If you can get arbitrarily close to a point, you can get to the point itself.
I guess I should be more technical and say that real analysis solves this problem, because what's really doing the heavy lifting here is the topology induced by defining a metric on ℝ^3 in combination with the notion of limits.
Clearly, not all infinite sequences can be summed. So e.g., 1, -1, 1, -1, … has no sum.
Now suppose that Achilles takes alternate forward and backward steps a (countably) infinite number of times. The first step takes one second, the second step takes half a second, and so on. (Each step covers the same distance.) Where does he end up after 2 seconds?
There’s no sensible answer to that question. Does that mean that Achilles can’t in fact traverse that particular sequence of steps? But then, why should he be unable to traverse a particular infinite sequence of steps merely because its sum is undefined? After all, the result of each individual step is perfectly well defined. If it’s possible in general to traverse infinite sequences, what stops him traversing that one?
To me, this just seems like Zeno’s paradox all over again. The mathematical treatment is more sophisticated, but the underlying paradox remains.
Zeno himself probably wouldn’t have distinguished carefully between summing an infinite sequence and spatially or temporally traversing it, since both notions would have seemed equally absurd from his point of view. Modern mathematics has shown us that the former isn’t in fact absurd. But Zeno’s paradoxes are arguably about the latter.
> Clearly, not all infinite sequences can be summed. So e.g., 1, -1, 1, -1, … has no sum.
Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.
More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.
>Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.
I am not sure what you mean here. You can calculate the sum of the time series, but you can't calculate Achilles' final position, which is the question at issue. The question remains: if it's possible in general to traverse an infinite sequence of steps in space, why is it not possible to traverse the one that I specified? "Solving" Zeno's paradox by admitting the possibility of traversing an infinite series of points in space or time seems to give rise to paradoxes just as deep as the originals.
> The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.
Yes, that was Aristotle's observation and a key part of his proposed solution to the paradox. The problem is that this explains why it's possible to sum the series, not why it's possible to traverse it. You seem to be taking the position that any series that cannot be summed cannot be traversed. But why should that be so?
Thinking about this a bit more, I think what I'm trying to say is that Zeno's paradox is more about supertasks than it is about the problem of computing the sum of an infinite series. There's a nice summary article that I found here:
They're not mathematically paradoxical, but that doesn't necessarily mean that the paradoxes are solved, because there's more than just math going on. A lot of the paradoxes hinge on the question of whether it is in fact possible to traverse an infinite series of positions in space or moments in time. I have no idea whether it is or it isn't, but the issue isn't settled by calculus. Calculus allows you to figure out what the result would be if such a traversal were to occur.