By David Siegel Bernstein, PhD
Welcome back to Part II of our discussion on infinity.
Drum Roll. Now as promised last month, I'm going to take a bullet in the name of math. My arch nemesis, who oddly enough volunteered for this job, aims her rifle at me and fires. According to math, do I get hit? (And yes, she is a good shot.)
If we believe Zeno of Elea (living during the 400s BC), I'm safe. Of course he didn't know about rifles, but he did like to devise devious paradoxes similar to this situation. The bullet flying toward me passes half way across the room, then half the remaining distance, then half again, and so on. I've divided its journey into infinite number of shorter and shorter segments. So in this case, Zeno would have argued that motion is an illusion.
The idea is (sort of) consistent with math because any line segment can be divided into infinite number of points. In the sad story of my shooting, the bullet must pass through an infinite number of points making me infinitely far away from the bullet. It's always getting closer and closer to me but never hitting me.
Math has saved me . . . well, except for when it doesn't. You see according to physics, and my foe's true aim, I'm going to get hit. Why?
I'll give you Aristotle's (384 BC – 322 BC) answer. Like Zeno, he didn't know about rifles, but he was aware of the halving paradox. He would have considered how long it took the bullet to reach me. Let's assume it's a very slow bullet, taking a full second to reach me and I'm slow to react. He'd argue that each interval of the series (all the halves) have a corresponding time. Assuming a constant bullet speed, as the distance gets shorter so does the time until impact. So I'm running out of time. All the time intervals added together can not exceed a second. Therefore it can not take an infinite amount of time to reach me.
So there is a limit (Do you recall limits from Part I? No? You could give it a re-read. I won't tell . . . I swear.) and it isn't infinity. If it took a half of a second to reach the halfway mark, then it can not be the case that the total time it takes to hit me to be greater than a second. You can't pass an INFINITE number of points in FINITE time.
So I've been shot (damn you physics). But don't worry it was only a glancing wound, I'll survive to write more.