Much has been written about Zeno’s paradoxes.
The link above states the paradoxes, as well as the various resolutions offered over the years, and the problems with some of the various resolutions.
We have a special guest with us for this article, Zeno himself. He’ll explain why Achilles will never catch the tortoise. Take it away, Zeno.
Zeno: Assume Achilles takes a step of length 1/2, then a step of length 1/4, etc. Will he actually ever touch the line in the sand one unit away from where he began? No, he never will. 1 is the Least Upper Bound of his steps. No matter how many steps he takes, even if he keeps taking steps until the Universe freezes over, he will never touch the line at distance 1.
Smiling Dave: Yes, you are right. But Achilles does not walk that way. He takes steps of the same length each time.
Zeno: Achilles does not actually walk that way, I agree, but we can analyze his walk, dividing it in our minds into an infinite number of steps. Then, when we try to put the infinite number of steps back together, we cannot, because Achilles is mortal and can only take a finite number of steps.
Same thing with my Arrow paradox. The arrow does not fly before our eyes through an infinite number of positions. It is our mind that analyzes the flight of the arrow as being made up of an infinite number of steps. And then comes the same punchline as with Achilles. Having cut the path up into an infinite number of steps, we cannot put it back together, because we have no way of adding an infinite number of things together.
SD: You understand infinity, Zeno, as well as anyone. But infinity is a very tricky thing. Two infinite sets A and B can be equal, even though A has everything B has, and more [Galileo’s Paradox]. A hotel with an infinite number of rooms, all of which are filled, can accommodate an infinite amount of new guests, such that there there is only one person in each room [Hilbert’s Grand Hotel]. The unit sphere, because it has infinitely many points, can be cut up into finitely many pieces, and put together into a sphere ten times the area [Banach Tarski Paradox]. And some infinite sets are “more infinite” than others [Cantor’s Diagonal Argument].
Zeno: Say what?
SD: All that has been proved by modern mathematics. It’s after your time.
Zeno; And your point is?
SD: Nobody has ever seen the infinite. We grasp it not by direct experience, but by mental processes. Since we have no experience to guide us, we have to be very careful when using this tricky thing, which can be proven to behave so paradoxically, as a model of reality. If our attempt at modelling the flight of the arrow as passing through an infinity of points, or of Achilles passing through an infinity of points, leads us to patently false conclusions, then all we have shown is that our model assumes something about infinity that is incorrect.
Zeno: Dave, that’s not good enough. Yes, infinity is tricky. But you have to show me exactly where my mistake is.
SD: Both your paradoxes assume you cannot take an infinite amount of steps in a finite amount of time.
Zeno: Correct. Wikipedia got it right, and I quote. Simplicius has Zeno saying “it is impossible to traverse an infinite number of things in a finite time”. This presents Zeno’s problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a “last event”?
SD: And who told you that you cannot take an infinite amount of steps in a finite amount of time?
Zeno: It’s common sense.
SD: Sorry, Zeno, you can’t use “common sense” when it comes to infinity. You either prove it from first principles, or you don’t. And you certainly haven’t proven it.
Zeno: I see that now, Dave. Thank you for clearing away millennia of confusion. What really brought it home to me was what you wrote about being able to fill an already full hotel, over an over, if there are an infinite number of rooms. That really shocked me, and I see now that you can’t make any unproven assumptions about infinity.
Dave: You are welcome. BTW, all of those supposed solutions over at Wikipedia are really silly.
Zeno: I know, right? They just don’t get it. For instance, what’s the point of running to quantum mechanics and a finite universe and discrete time and all that other nonsense. That’s just throwing in the towel, admitting Newtonian physics is logically flawed.
And those guys you see on Youtube, who think they discovered America because they can sum an infinite series on paper. How badly they missed the boat! Just because an infinite series has a sum, that doesn’t mean Achilles can take an infinite number of steps.
SD: I thought you died before Newton.
Zeno: You have anything about my other arrow paradox?
SD: Which one is that?
Zeno: Here’s Wikipedia about it:
In the arrow paradox Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.
Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.
SD: Newton solved that with his concept of instantaneous velocity.
Zeno: Oops, missed that one. I died before his time, you know.