Welcome to the Great Athenian Man-Tortoise Run-off. My name's Zeno and I'll be your commentator for the big race. I have to say, however, that the result is a foregone conclusion. Achilles has made the terrible mistake of giving Tarquin the tortoise a 100-yard head start. Let me explain.
Tarquin's tactic is to keep constantly moving, however slowly. If Achilles is to overtake Tarquin, first he must get to where Tarquin is when the race starts. That will take him several seconds. In that time, Tarquin will have moved on a little and will then be a short distance ahead of Achilles. Now if Achilles is to overtake Tarquin, he must again get to where Tarquin is first. But in the time it takes Achilles to do that, Tarquin will again have moved forward slightly. So, Achilles once more needs to get to where Tarquin is now, in order to overtake him, in which time Tarquin would have moved forward. And so on. You get the picture. It's logically and mathematically impossible for Achilles to overtake the beast.
Still, it's too late to place your bets on the tortoise now, because they're under starter's orders, and...they're off! Achilles is closing...closing...closing...Achilles has overtaken the tortoise! I can't believe it! It's impossible!
Baggini, J., The Pig That Wants to Be Eaten, 2005, p. 46.
We know this isn't impossible, but what's going on here? First, like a good engineer, let's set up all the facts and assumptions and do some basic math. The fastest human speed ever recorded is 27.78 miles per hour (during Usain Bolt's 100-meter dash), so let's say Achilles can run 25 mph. The fastest recorded speed for a tortoise is 5 mph, so let's use that nice round figure for Tarquin. We'll ignore the short period of acceleration during the start of the race just to make the math much easier. So since rate x time = distance, we can calculate that Achilles (running at 12.22 yards/sec) would overtake Tarquin (running at 2.44 yards/sec) in 10.2272 seconds when they would both have gone 125 yards. After that, Achilles would steadily pull away until the end of the race. This is the straightforward way of looking at this problem. But what about Zeno's way?
Zeno's method of describing Achilles as always having to catch up to Tarquin relies on looking at a series of ever smaller steps and claiming you can't get over them all. To see just how small these steps get though, let's run the math. Using the same rates of speed described above, we get:
- Step 1: Achilles goes 100 yards in 8.18 seconds, while Tarquin makes it 20 yards.
- Step 2: Achilles goes 20 yards in 1.64 seconds, while Tarquin makes it 4.0 yards.
- Step 3: Achilles goes 4.0 yards in 0.33 seconds, while Tarquin makes it 0.8 yards.
- Step 4: Achilles goes 0.8 yards in 0.07 seconds, while Tarquin makes it 0.16 yards.
- Step 5: Achilles goes 0.16 yards in 0.01 seconds, while Tarquin makes it 0.032 yards.
- Step 10: Achilles goes 0.0000512 yards in 0.00000419 seconds, while Tarquin makes it 0.0000102 yards.
In just 5 steps, we get to the kind of time gap seen in an Olympic photo finish. In 10 steps we get to a gap that is 4 millionths of a second -- far beyond anything we are capable of seeing with the naked eye. Even with our best tools, the current world record for the shortest unit of measured time is 12 attoseconds (1.2 × 10−17 seconds). Zeno crosses that threshold in just 27 steps. What about the next million steps?
Some philosophers have attacked Zeno's paradox this way by saying the universe isn't actually infinitely divisible. At some point in this process, you get to a step that must be the size of the fundamental grain of space-time that makes up the fabric of the universe. After that, Achilles and Tarquin must cross these steps at differing rates until Tarquin is passed. But we don't actually know if the universe is infinite or finite (see Planck lengths for a brief discussion), so this doesn't conclusively refute Zeno.
What all this math makes clear to me, however, is the way that Zeno is actually refuted. We see quickly how Zeno is relying on slowing down time to make smaller and smaller observations during the moments when Achilles is about to overtake Tarquin. As long as we give Zeno this power, his paradox holds true. But time waits for no man. During the one-hundredth of a second between 10.22 and 10.23 seconds, Achilles will pass Tarquin, and no amount of thinking about this will make that time stand still. Achilles moves independently of Tarquin and his location in the universe will be farther along the path of the race. The concept of infinity is only a mathematical abstraction so there is no real-world power in saying someone must overcome "an infinite number of obstacles." The barriers are always numbered and we see they can be surmounted.
That's 16 thought experiments passed for me. Only 84 more to go...