I understand that Bertrand Russell, in repsonse to Zeno's Paradox, uses his concept of motion: an object being at a different time at different places, instead of the "from-to" notion of motion.

I also understand that this concept solves Zeno's Paradox of the arrow, as his concept aptly describes the motion of the arrow; however, his concept of motion does not aptly resolve Zeno's Paradox when told as the "dichotomy" and the "race". However, I don't understand the following quote:

But it is not essential to the existence of a collection, or even to knowledge and reasoning concerning it, that we should be able to pass its terms in review one by one.

Can someone explain why this addition to his notion of motion not fully account for the "race" and the "dichotomy"?

  • Dear Curious_Mind- A segment is not a distance; Is that solved? CMS – user37981 Apr 12 '20 at 12:57
  • I'm confused with what you are trying to imply. Could you please elaborate? – Curious_Mind Apr 12 '20 at 13:09
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    By saying that an arrow travels halfway, the so-called paradox converts a distance into a segment. Halfway is not a distance traveled, it is a portion of the distance measured as a fraction. Reducing the halfway 'segment' until it disappears has nothing to do with distance. It is the wrong category of measurement to use when suggesting that motion is impossible. Cheers CMS – user37981 Apr 12 '20 at 13:18
  • Be(r)trand who ? – Mauro ALLEGRANZA Apr 12 '20 at 13:28
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    According to whom does his concept of motion "solves Zeno's Paradox of the arrow", but "does not aptly resolve Zeno's Paradox when told as the "dichotomy" and the "race""? Since you are asking I assume you read it in a commentary somewhere. Since Russell objects to counting milestones one by one before passing a collection of them is allowed this should work for dichotomy as well. The problem with Russell's "solution" is that his concept of motion is a concept of a collection of rests, and not of motion at all. – Conifold Apr 13 '20 at 2:32

Zeno's paradox should be understood as an inadvertent reductio ad absurdum.

That is, the paradox shows that if moving from A to B required, as is implicitly assumed in the paradox, that we should be able to "pass its term in review one by one", that is, that we should be able to review one by one all successive locations in an uncountable continuum of locations, then moving would indeed be impossible. Yet, as the paradox itself in fact assumes explicitly this time, and more importantly as we can all check for ourselves, moving is not only possible but happens and happens all the time. Therefore, the paradox's implicit assumption is falsified.

Zeno's argument only works as a paradox because the crucial assumption, i.e. that counting an uncountable set is somehow a necessary pre-condition for moving, is not articulated. Thus, the only assumption which is falsifiable doesn't appear and isn't so easy to uncover. There is nothing in evidence which is falsifiable.

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    That's not true. 1/2, 1/4, 1/8, 1/16 ... (in one version) or 1/2, 3/4, 7/18, 15/16, ... (in another) are countably infinite sets. – user4894 Jan 8 at 18:31
  • However there is an uncountable set of positions through which the arrow must pass on its journey, being the irrational number in the range zero to (length of journey), so there is still an argument to be made. Whether it’s any more persuasive than for a countable set is another matter. – Frog Jan 9 at 1:53
  • @user4894 Read again. I said "uncountable continuum of locations". – Speakpigeon Jan 9 at 16:41
  • Zeno's argument doesn't seem to required that. Suppose the line is made up of only rational points. The argument still works. Suppose it only contains the dyadic rationals, the ones of the form n/2^m. The argument still works. – user4894 Jan 9 at 21:05
  • user4894 "Countably" infinite sets? Words are really cheap. So, go on, count 1/2, 1/4, 1/8, 1/16 till the end and come back when you're done. – Speakpigeon Jan 10 at 11:36

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