# Is there an answer to Zeno's paradox of Achilles and the tortoise ?

Is there an answer to Zeno's paradox of "Achilles and the tortoise"?

I'm not sure if there is an answer or way to resolve this paradox nor do I know if it would go in the category of physics or math (even though I've heard that as a math problem it was solved a few centuries ago).It goes something like a tortoise runs ahead of Achilles but it is Achilles who is always behind it and ALWAYS only makes it as far as the last spot where the tortoise was no matter how infinitemestal..:uhh:..or something to that effect.I'm sorry to say that I cant really explain to well.There's a link below that'll explain it better

http://en.wikipedia.org/wiki/Achilles_and_the_Tortoise#Achilles_and_the_tortoise

My question is though has there been in a way to mathematically solve it and/or is there a way to solve it at all?.

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Yes, using elementary calculus. The link you posted contains an explanation. The problem reduces to a convergent infinite series.

If you want an in-depth exlanation, "Everything and More: the compact history of infinity" explains what the exact problem is, and how it was finally solved. It's a book and requires about two semesters of calculus to grasp all of the concepts in the book, though you can get through it with less.

It does indeed simplify down to a convergent infinite sum, but in order to get there, there was a whole theory (and in George cantor's mind philosophy) of infinity needed in order to proof definitively, without a doubt, Achilles really can pass the hare, that arrows really do move, and one can indeed cross the street/touch their nose with their hand.

The book has lots of math in it, but is written extremely well and is entertaining.

The basic idea is this: suppose for the tortoise to reach the first point takes 1 second. Of course, during that time Achilles has gone even further. To get to that new point takes 1/2 seconds. The next stage takes 1/4 seconds. Etc. This goes on ad infinitum, so goes the paradox.

But how much time does it take the tortoise? Well $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ seconds. This sum is equal to 2 seconds. (*)

(*) to see why, note that if you divide every term by 2 and you add 1, you get the same sum, hence $\frac{\textrm{sum}}{2} + 1 = \textrm{sum}$. Hence $\textrm{sum} = 2$.

Actually, the reasoning in my last post was a bit of cheating. At least the part where I said it was finite and equal to 2 was cheating. After all, the same reasoning would gives that $1 + 2 + 4 + 8 + \cdots$ is -1, since if you multiply the sum by 2 and add 1, you get the same sume. I.e. $2 \textrm{sum} + 1 = \textrm{sum}$ such that $\textrm{sum} = -1$.

A more rigorous way to prove that $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$ goes as follows: if you take only a finite number of terms (say N+1), i.e. $\textrm{sum} = 1 + \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^N}$, then one can note that $2\textrm{sum} - 2 + \frac{1}{2^N} = \textrm{sum}$. Solving this gives that $\textrm{sum} = 2 - \frac{1}{2^N}$. If we let N get bigger and bigger (i.e. we take more and more terms into our sum), we see that the sum approaches 2 (since $\frac{1}{2^N}$ becomes smaller and smaller).

Hurkyl
Staff Emeritus
Gold Member

.It goes something like a tortoise runs ahead of Achilles but it is Achilles who is always behind it and ALWAYS only makes it as far as the last spot where the tortoise was no matter how infinitemestal..:uhh:
Achilles is indeed always behind the tortoise during the time period where he's catching up to it. What's the problem with that?

Well thanks for the replies guys but an alternative for a laymen like myself..:uhh:.is there like a philosophical way to solve this paradox.

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Hurkyl
Staff Emeritus
Gold Member

Well thanks for the replies guys but an alternative for a laymen like myself..:uhh:.is there like a philosophical way to solve this paradox.

Before one can 'solve' a paradox, one must first know what the paradox is. What contradiction do you think arises? Explain in precise detail.

One can 'philosophically' 'solve' a paradox by observing that the claimed paradox is lacking a valid argument.

e.g.
Achilles who is always behind it
why would you think Achilles is always behind the tortoise, rather than only being behind the tortoise during the time period when he's catching up?