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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.

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.

SD: Which one is that?

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. – as recounted by Aristotle, Physics VI:9, 239b5

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.

1. Lio says:

Dave,

Is there still a debate about the zeno’s paradox nowadays? Zeno’s paradoxes have been solved by the limit concept long ago, right? Will you abandon economic issues now for math problems? Yet there are many bad blogs in Economics that should be corrected!

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2. Smiling Dave says:

According to Wikipedia on Zeno’s Paradox, there is indeed still a debate.

The concept of limit does not solve the basic problem Zeno raised. It gives a way of calculating an answer, but that’s not what Zeno was asking. I refer you to the Wikipedia article. His essential question was how can you do an infinite amount of tasks in a finite amount of time? The concept of limit doesn’t answer that.

Nor do the other solutions proposed by the folks Wikipedia quotes there understand what the problem really is. Most of them think they have answered the question by resorting to new models of reality [it’s discrete, it’s space-time, etc.], which again shows ignorance of the essence of the problem, to wit, do we have to dump the math that models Newtonian Physics, aka the only math we have, because it contains a paradox? Zeno wasn’t challenging our view of reality, but our rules of logic.

I haven’t abandoned economic issues, although I have a feeling I must learn a lot more to be able to contribute further. The internet, and the world in general, is indeed full of economic ignorance, but I don’t feel inspired to say much more about it.

If you or any reader has a particular question or topic they’d like to see talked about, I’ll do it if I can.

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3. Lio says:

Dave,

In terms of pure logic, Zeno’s paradoxes are not paradoxes at all if you’re familiar with specific tools of mathematics. If you have a little time for that, I suggest you take a look at a math book on this topic and in particular calculus and infinite geometric series.

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4. Smiling Dave says:

I’m familiar with calculus and with infinite geometric series. Let me quote from http://web.archive.org/web/20100418141459id_/http://www.philosophers.co.uk/cafe/paradox5.htm :

One common reply is that Zeno has misunderstood the nature of infinity. Modern mathematics, it is said, has shown that the infinite sequences that Zeno generates do have a finite sum. In particular, to take the Racecourse example, the sequence 1/2 + 1/4 + 1/8 + 1/16 + . . . is equal to 1.

This reply, however, misunderstands what modern mathematics has shown. Mathematicians do use sequences such as 1/2 + 1/4 + 1/8 + 1/16 + . . . but they say that they have a limit of 1, or tend to 1. That is, we can get nearer and nearer towards 1 by adding on more and more members of the sequence, but not actually arrive at 1 – this would be impossible because we are considering an infinite sequence. So far from providing an argument against Zeno, mathematics is actually agreeing with him!

Further, this reply seems to miss the point of Zeno’s argument: simply pointing out that there is a branch of mathematics that deals with the infinite does not reduce the puzzling aspects of the Paradoxes. We know that races can be run and that Achilles will overtake the Tortoise, what we want to know is what is wrong with the arguments that show that these things can’t happen.

That nails it exactly!

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5. Lio says:

Dave,

The problem is that you consider that a limit can never be reached. A limit is a finite number which equals the sum of n terms of a series when n is infinity. The sum of n terms of your geometric series is really 1 when n is infinity! The zeno’s paradoxes can not be solved if you consider that no limit can be reached.

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6. Smiling Dave says:

A limit is a finite number which equals the sum of n terms of a series when n is infinity.

Modern mathematics does not see it that way, Lio. You’ll have to ask a math professor, because the topic is too intricate to discuss in this venue.

The zeno’s paradoxes can not be solved if you consider that no limit can be reached.

There is a solution, the one I proposed.

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7. Lio says:

Ok Dave,

I must concede that I am not a native english speaker which can explain why my defnition of a limit is not very clear or appropriate.

I’ll try to do better now giving you proof that the sum of the geometric series s (1/2 +1/4 +1/8 + …) is exactly 1:

If you multiply through by 1/2, then the initial 1/2 becomes a 1/4, the 1/4 becomes a 1/8, and so on:

1/2s = 1/4 +1/8 +1/16 +…

Subtracting the new series (1/2)s from the original series s cancels every term in the original but the first:

s – 1/2s = 1/2, so s = 1!

When you think a little about this result, you can realize that it is not so strange after all. If we can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give us back the finite distance we started with.

I hope I have now made it clear.

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8. Smiling Dave says:

Lio,
1. The question never was “What is the sum of the geometric series?” Of course it is one.
The question was an entirely different. Wikipedia explains:
Zeno’s arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. 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”?

2. As I wrote in the article, when we divide up a finite distance into an infinite number of small distances, we are doing that in our heads. There are no physical tools or steps involved. But when we add all those distances together, we are doing it not just in our minds. There is a human, Achilles, on a race track taking a step at a time and he does not live forever. He can only take one step at a time. We can divide things and add them in our minds instantly, but he has to actually walk the steps in real time. And just because we divide it in our minds does not mean Achilles can add it back together with his walking.

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9. Lio says:

Dave,

?!? You seem not to realize that I have already answered these questions. Never mind. We can not be agree all the time!

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10. Redmond Wallace says:

This is a simple rough draft of another argument on the matter. http://danifestdestiny.blogspot.com/2014/02/zenos-paradox.html

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11. Smiling Dave says: