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Clarified paragraph beginning "even if one accepts this solution".
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Ludwig V
Ludwig V
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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, sobut 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.

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. Yet we know that Achilles can, 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.

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.

Source Link
Ludwig V
Ludwig V
  • 3k
  • 1
  • 6
  • 25

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. Yet we know that Achilles can, 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.