So, Zeno assumes that, to go from the mark at $1m$ to the mark at $2m$ we've to do an infinite number of tasks. Like the task of getting to $1.001m$, the task of getting to $1.000005m$,the task of getting to $1.658m$,etc.

So, to perform an infinite number of tasks, one must take an infinite amount of time. I think that's where the loophole is. Well, yes, that would be true only if each of these tasks took a finite time to complete. If each of these tasks took $1s$, then, yes, it'd be true that completing all these tasks would take an infinite number of seconds.

But, the truth is that none of these tasks take a finite time to complete. If we divide the objective of getting from$1m$ to $2m$ into an infinite number of tasks, and let $dx$ be the infinitely small distance, then the infinite number of tasks are:

  1. Going from 1 to 1+dx

  2. Going from 1+dx to 1+2dx...................

$n-1$. Going from 2-2dx to 2-dx

$n$. Going from 2-dx to 2.

Well, none of these tasks take a finite time to complete. The time taken is also infinitely small. In each of these tasks, the distance we've to cover is $dx$ and hence the time taken is the infinitely small, $dt=\frac{dx}{v}$, if our speed is $v$. And, an infinite number of infinitely small times add up to give a finite time required to get from $1m$ to $2m$ like this: $$\int_1^2\frac{dx}{v}=\frac{2-1}{v}=\frac{1}{v}$$ So, I don't see any paradox. Where's the paradox?

I don't understand why people introduce concepts like quantization of space and time to explain Zeno's paradox.

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    $\begingroup$ One problem with your argument is that there are no "infinitely small" quantities in the standard real numbers. Another problem is that even if there are, there's no proof that there's any such thing in the physical universe. Zeno is not talking about math, he's talking about the world. Modern physics tells us that it's meaningless to talk about intervals of time below the Planck length. You can't reconcile your theory with either physics or standard math. It's true that there are systems of math with infinitesimals, but they won't help your argument. $\endgroup$ – user4894 Feb 25 '17 at 4:33
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    $\begingroup$ "But, the truth is that none of these tasks take a finite time to complete." Every task he talks about takes a finite time to complete because each task is to travel a finite distance, which takes a finite amount of time to do. $\endgroup$ – Chai T. Rex Feb 25 '17 at 4:39
  • $\begingroup$ @user4894 The only way to divide going from $1m$ to $2m$ into an infinite number of tasks is to assume that each task is infinitely small. For example, if each task is to cover a finite distance, say 0.01m, and there are an infinite number of such tasks, then the total distance we've to move will be infinite. So, if infinitely small distances don't exist, then there's no way to divide a distance into an infinite number of tasks, and therefore the paradox doesn't exist. $\endgroup$ – Dove Feb 25 '17 at 4:40
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    $\begingroup$ You can in fact have infinitely many finite distances adding up to a finite distance. For example, $\dfrac 12 m + \dfrac 14m + + \dfrac 18m + ... = 1m$. None of the distances are infinitesimal, they each have a length. But they get smaller and smaller fast enough so that even infinitely many of them will get you a finite distance. You can also substitute time instead of distance throughout my whole comment. $\endgroup$ – Ovi Feb 25 '17 at 4:51
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    $\begingroup$ That series was in response to you saying that if you add up infinitely many distances, you neccesarily get an infinite distance. $1$ being a limit shows that even if you add up infinitely many terms of that sequence, you get something which is less than or equal to $1$, which is still a finite thing. $\endgroup$ – Ovi Feb 25 '17 at 5:02

Zeno's paradox is fundamentally a philosophical problem. Many philosophy problems can be approached mathematically, but you have to be careful to justify that you're using the right math. In your solution, you're choosing to model time in a certain mathematical way. That needs to be justified. You could also (as you mention) solve the problem by modeling time and space with the integers and just asserting that there's only finite units of both, so the paradox doesn't arise.

However, doing math in these models isn't a complete philosophical argument. You need to argue that time is, in fact, like the real numbers in the relevant way. Otherwise someone can just say "well, in your model your idea works but in the real world it doesn't because the world doesn't look like your model."

There are many ways in which Zeno's paradox can fail. The question is which one is actually the way it can fail.

  • $\begingroup$ Great answer. So, none of the explanations can be the actual explanation? $\endgroup$ – Dove Feb 25 '17 at 5:22
  • $\begingroup$ @Dove Mathematics /by itself/ cannot be. If you can present an argument (presumably grounded in physics) as to why your model describes the world, that together with the math is enough. This turns out to be surprisingly hard to do, which is why people often resort to quantized space. It might not be necessary to understand why Zeno's Paradox fails, but it's the easiest way to give a justified explanation for the fact that it does. $\endgroup$ – Stella Biderman Feb 25 '17 at 5:24

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