Assignment 2. Zeno's Paradoxes of Motion

For submission

1. In the dichotomy, a runner must run half-way along the course, and then half-way again, and so on indefinitely. If these infinite runs could be completed, the runner would have completed the course. What happens if we replace the "half" with a "third"? That is, to complete the course, the runner must run a third of the way, and then a third of the way that remains, and so on indefinitely. (Note "... a third of the way that remains ...")

Immediately after the runner has completed all these third-point runs, where is the runner? At the end of the course? Or somewhere along the course?

To answer, consider the distance remaining at each stage. (The figure shows these distances for the original paradox.)

2. In question 1, replace "third" with "one tenth." Again:

Immediately after the runner has completed all these tenth-point runs, where is the runner? At the end of the course? Or somewhere along the course?

3. In the Achilles, have Achilles and the tortoise switch positions. Achilles starts ahead of the tortoise and the tortoise chases after him. To catch Achilles, the tortoise must move to Achilles' starting point, and then to the position to which Achilles moved, and so on indefinitely through infinitely many "catch up points."

In the original Achilles, this reasoning did not establish that Achilles must fail to catch the tortoise. Does this reasoning now work in the switched version? If so, what has changed?

For discussion

Not for submission

A. In question 3 above, imagine that the race is held on an infinite track and allowed to run for all time. In that case, over the infinity of time of the race, the tortoise will eventually arrive at every one of the "catch-up points." Does it now follow that the tortoise catches Achilles?

B. There are three resolutions of the dichotomy in the chapter:
     "potential and actual infinity,"
   "summing an infinity" and
     "completing an infinity."
How does each work? Which of them do you prefer?

C. For those who know calculus, is the calculus-based response to the arrow better than the one in the chapter?

D. The stadium does, in my view, need more sophisticated ideas for its resolution than those presented so far. Or does it? Can you resolve the paradox?