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Solution to the Achilles/turtle puzzle!

sorgoth

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I've been thinking about this for a while, and I think I've figured out why achilles never actually overtakes the tortoise by most calculations.

It's because we're thinking of Achilles and the tortoise as points, not as figures that take up space. Two points can always get closer to each other without actually touching, but two objects, once you get close enough, actually touch. So Achilles does catch the tortoise!
 
I don't know if your "taking up space" logic is necessarily the right method to find the answer. From my understanding of the problem, even if achilles and the turtle were points achilles would still eventually win. The solution lies in the fact that you are thinking of a limited number of steps, in which case achilles would never win. However, it is the case in mathematics(and in this problem) that an infinite number of steps can add up to a finite answer. For example 1/2+1/4+1/8.... = 1. My apologies if this doesn't make much sense, reading back, it doesn't make sense to me anymore.
 
I think Achilles and the tortoise are supposed to be thought of as points. Achilles still catches the tortoise because although you can split their movements up into an infinite number of steps until they get to the same point, the infinite number of steps only take a finite time to complete, so he still catches the tortoise.

It's like an infinite sequence where f(n+1)=f(n)/2. Assuming f(1)=1, the total of the infinite sequence is 2, a finite number.

Or where you just being silly? :cool:

David

PS Argh! Gazumped!
 
I assume you're talking about Zeno's paradox. This is no mystery and it's not a paradox, it just seems like one.

Basically the argument is that Achilles can never reach the tortoise because he would have to first get halfway there, and he can't do that because he would first have to get 1/4 of the way there, and first he would have to get 1/8 of the way there, on and on, the argument being that this adds up to an infinite distance that Achilles can never traverse.

The error is this: An infinte series does not have to add up to infinity. Many infinite series converge, meaning they "add up" to an actual number. In this case all the infinte fractions add up to 1, not infinity, and Achilles can easily pass the turtle.

Edited to add: Oops, I'm a little behind the times...
 
sorgoth said:
So 1+1/2+1/4+1/8...=2?

...How?
Click to expand...

It's easier if you try to understand 1/2 + 1/4 + 1/8... = 1. You can see the result getting closer and closer to 1, the more terms you add. At an infinite number of terms, the series equals one.

To understand it any more you'd have to study infinite series.
 
Or you could just mumble something about space-time not being infinitely divisible? :cool:
 
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