Interview with Paradox

[myndzi]

That doesn't necessarily have to be the misunderstanding. Say someone wants to travel 1 mile. Normally, with a finite number of steps, you can describe how he got there easily. "He took 2000 footsteps. First he took footstep 1, then 2, (...), then after footstep 2000 he arrived at his destination.

With an infinite number of steps, you can't do this. Yes, the sequence 1/2, 3/4, 7/8, 15/16, (...) approaches the limit of 1. There's no problem with summing an infinite number of partial miles into a finite 1 mile.  However, there is no particular step in that sequence that comes right before the destination. There is no point, while the act is taking place, where there isn't at least some infinitesimal gap between it and the destination. I believe this is what gives most people problems with this paradox. It is not satisfactory to simply acknowledge that, by a mathematical definition, he traveled a mile.

What he's getting at is that the perspective to take involves time, too. When you say that the person will 'never' reach the destination, you're involving time. If it takes 10 seconds to walk across the room, for example, you have to realize that when you take 1 1-millionth of a step it takes 1 1-millionth of the amount of time, too. So, never reaching the destination goes hand in hand with never having 10 seconds elapse.

This seems to relate in a useful way to the .9 repeating = 1 thing. They have the same meaning, but are expressed in different ways. Equally, walking X feet in Y seconds has the same meaning as "approaching the limit of X feet in the limit of Y seconds" -- you can divide it infinitely as long as you can perform the math, but it'll still reach the goal at some point, unless you can figure out how to stop time!

[caffeine]

Once again, the problem isn't necessarily whether it can be done in a finite time/length. It has to do with how things break down logically once you look at the steps closely.

It may be easier to see what I mean by looking at a similar problem: Black's infinity machine. There is a machine that takes a ball to point A at 1/2 an hour, point B at 3/4 an hour, back to A at 7/8 an hour, and so on. (Keep in mind that we're trying to see if this is possible logically, not physically.) What point will the ball be on at the end of that hour? The machine performed an infinite number of tasks in a finite time-- no problem there. The problem comes when you think about how, since he's performed an infinite number of tasks, he will always follow point B with A, and he will never go to point A without that being followed by B. In other words, you cannot logically deduce where the machine will end up by following the task as it is happening. The same can be said for the dichotomy paradox. You're never able to determine just where and when the runner is right before he arrives at his destination.

[Katatoniopeth]

Wow, I never expected this thread to get so deep! I am looking forward to doing more interviews in the future, this is exciting