Final Steps and Zeno's Paradox

Question

In the SEP article on supertasks, it states that:

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence. The existence of a final step was similarly demanded on a priori terms by Gwiazda (2012). But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy. From Black’s argument one can see that the Zeno Dichotomy cannot be completed in the first sense. But it can be completed in the second. The two meanings for the word “complete” happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.

The question is a relatively simple one, that being, if the journey is from point A to point B, and the completion of the journey entails reaching point B by traveling point by point, how can one complete "every step" if reaching point B is one of the steps (as the reaching of point B would be a "final action")? In what sense can you posit completion of all the steps if it does not include the crossing of the point that signifies completion?

Answer 1

I think the simple answer is that the SEP article is wrong.

The argument seems to be:

1. The infinite sequence 100m, 150m, 175m, ... has no final term
2. The infinite sequence 100m, 150m, 175m, ... approaches 200m
3. One can walk 100m, then walk 50m, then walk 25m, ...
4. Therefore one can walk 200m without walking some final distance

The first thing to note is that this argument is invalid. The second premise only says that the sequence approaches 200m, whereas the conclusion is that one reaches 200m.

The second thing to note is that if we treat each distance in the infinite sequence as a single footstep then it isn't clear what it means to have "completed" every step in the sequence. At what distance is my foot after every step in the sequence has been performed? It certainly isn't on or past the 200m finish line because such a footstep is not a term in the sequence.

Answer 2

Your first question is:-

if the journey is from point A to point B, and the completion of the journey entails reaching point B by traveling point by point, how can one complete "every step" if reaching point B is one of the steps (as the reaching of point B would be a "final action")?

If point B is one of the steps, then a next step can be defined, so point B cannot be the final action. The final action is not defined, and arguably the possibility of such a thing is excluded by the definition of the problem.

Your second question is:-

In what sense can you posit completion of all the steps if it does not include the crossing of the point that signifies completion?

The passage you quoted does suggest there is another sense of "complete". I think it is this:-

On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy.

Working out what this means in the SEP article is not easy, but I'm pretty sure that it is referring to the demonstration (long after Zeno) that a convergent series of this kind does have a sum, on which the series converges. Clearly, this solution has not convinced many other philosophers and mathematicians. The problem is that "converge" does not mean "reach" or "complete". The so-called "sum" is a limit, which the series can approach, but not reach.

Even if one accepts this solution, it is surely clear that the series cannot pass its limit and that is what is implied by your phrase "crossing the point that signifies completion". Yet we know that Achilles can not only reach the finishing lime, but pass beyond it. So the solution is, to coin a phrase, not complete.

What is hard to grasp here is that one can complete every step in the series, but it does not follow that one can complete all the steps. Each step is defined with perfect clarity and can therefore be completed. So any finite number of steps can be completed. But it does not follow that all the steps can be completed.

In any case, the SEP article explains another version of this problem:-

In this description of the Achilles race, we imagine winding time backwards and viewing Achilles getting ever-closer to the starting line (Figure 1.1.2).

There is no initial step in this task, so Achilles not only cannot complete his race, but cannot even start it.

Even if one accepts the sum of a convergent series as a solution, supertasks have been devised which evade it. Indeed, the SEP article goes on to consider a number of variants, each of which evades one or other possible solution. This does suggest that the sum of a convergent series does not get to the heart of the problem.