Looked for old threads on this, but none that I found were asking the exact question.
A long time ago, my French teacher (don’t ask why) pointed out the following paradox to my class.
If we believe space is infinitely divisible, then… a bullet fired at me from ten feet away will at some point be five feet away, then two and a half feet away, then one and a quarter feet away, then 7.5", then 3.75", then half of THAT distance, and so on.
If the distance between the bullet and my left nipple can be divided over and over again, how does the distance reach zero? In other words, how can the bullet make contact if there is always a tinier space available between us?
The issue here is that while the distance is halved, the time required to cover that distance is also halved, and that makes the sum of times and distances finite instead of infinite. I’m sure one of our resident mathematicians will be along shortly to explain this far more clearly than I have.
This is “Zeno’s paradox”. I’m sure we’ve done many threads on it.
The math squadron will be in soon to give solid rigor to the answer, but the simplest explanation in everyday language is that we have learned since the time of the Greeks how to handle infinite sums. This particular infinite sum reaches a limit of 1, or the distance between you and the bullet. The development of calculus depends on this understanding.
For starters, here’s the Wiki page on Zeno’s paradoxes.
Short version: You’re disregarding the fact that the TIME required for the traversal of each subsequent half-distance is also halved. So, assume the first 5 feet take 0.01 seconds (just a number, don’t nitpick with actual ballistics, please!); the next 2.5 feet take the bullet 0.005 seconds to cross, etc…
So you have the time required for the bullet to reach your chest = 0.01(1 + 0.5 +0.25 +0.125…) ; it just so happens that, using some basic calculus, the terms inside the parentheses can be proven to converge to 2; so you’d be hit after 0.02 seconds in this case.
Just for laughs, you should have asked your professor if he would volunteer to stand in front of a loaded gun and have it fired at him. After all, if he’s right, he can’t be hit…
The paradox is based on the notion that the sum of an infinite number of non-zero times must itself be infinite. This seems reasonable, but is in fact incorrect.
No, the point of the paradox and its solution is that an infinite number of sums must be added and accounted for. Zeno could have handled finite sums quite easily. But it took a new way of looking at the world for mathematicians to embrace infinities.
