I got critical feedback on the example I came up with in this post and wrote this blog comment in response:
I think I got mixed up in the following way:
I used an everyday example that varied in some details from the Achilles-tortoise race to check my understanding of Feynman’s explanation of Zeno’s paradox. I imported some background assumptions about how bills work (like lots tend to be regularly repeating on say a monthly basis) but then made the bills deviate from how bills actually work (and also made income deviate from how it actually works) in order to try to fit the ratios of the Achilles-tortoise race. So I basically did a poor rewrite of the original Zeno’s paradox using some superficial elements from the paradox in a new hypothetical where they didn’t make a lot of sense.
original post below
From one of the Feynman lectures on physics:
Even though we know roughly what “speed” means, there are still some rather deep subtleties; consider that the learned Greeks were never able to adequately describe problems involving velocity. The subtlety comes when we try to comprehend exactly what is meant by “speed.” The Greeks got very confused about this, and a new branch of mathematics had to be discovered beyond the geometry and algebra of the Greeks, Arabs, and Babylonians. As an illustration of the difficulty, try to solve this problem by sheer algebra: A balloon is being inflated so that the volume of the balloon is increasing at the rate of 100 cm³ per second; at what speed is the radius increasing when the volume is 1000 cm³? The Greeks were somewhat confused by such problems, being helped, of course, by some very confusing Greeks. To show that there were difficulties in reasoning about speed at the time, Zeno produced a large number of paradoxes, of which we shall mention one to illustrate his point that there are obvious difficulties in thinking about motion. “Listen,” he says, “to the following argument: Achilles runs 10 times as fast as a tortoise, nevertheless he can never catch the tortoise. For, suppose that they start in a race where the tortoise is 100 meters ahead of Achilles; then when Achilles has run the 100 meters to the place where the tortoise was, the tortoise has proceeded 10 meters, having run one-tenth as fast. Now, Achilles has to run another 10 meters to catch up with the tortoise, but on arriving at the end of that run, he finds that the tortoise is still 1 meter ahead of him; running another meter, he finds the tortoise 10 centimeters ahead, and so on, ad infinitum. Therefore, at any moment the tortoise is always ahead of Achilles and Achilles can never catch up with the tortoise.” What is wrong with that? It is that a finite amount of time can be divided into an infinite number of pieces, just as a length of line can be divided into an infinite number of pieces by dividing repeatedly by two. And so, although there are an infinite number of steps (in the argument) to the point at which Achilles reaches the tortoise, it doesn’t mean that there is an infinite amount of time.
So the error is reasoning from the infinite divisibility of a particular amount of time in the abstract to the conclusion that there is infinite time in reality. Is that right?
At this point I think I’m not really “solid” on this idea so I want to CHEW it some.
Using your mind you can conceptually cut up a length of time as many times as you want, but it’s still the same total length.
It’s kinda like if you had $100, and you reasoned as follows:
When I get my next bill for $100, I’ll also get another $10 of income.
When I get a bill after that for $10, I’ll also get $1 of income.
When I get a bill after that for $1, I’ll also get 10 cents of income.
And on and on.
I will never run out of money!
In reality, you won’t have infinite time to keep getting the smaller amount of income that lets you handle the smaller bills. At some point it’ll be next month, and the big bill will be do again, and you’ll be in trouble.
Likewise with the Zeno stuff. The tortoise doesn’t have infinite time to keep getting another increment ahead. Eventually he runs out, and Achilles laps him, and that’s that.