That said, I've always understood Zeno's paradox in what you might call the opposite way: as an argument NOT that Achilles will never reach his goal, but only approach it asymptotically; but rather, that Achilles cannot even begin moving.
It's basically a reductio ad absurdum of the assumption that space is comprised of an actually infinite number of discreet points.
Zeno's saying, look, if that were true, then in order for Achilles (or anyone, for that matter) to get anywhere - i.e. any arbitrarily distant point from his starting point - he would first have to get halfway there. but before he could get halfway there, he would first have to get halfway to the halfway point. and so on, ad infinitum. and because an infinite number cannot ever be traversed by incremental addition, Achilles would be stuck at the starting line of the race, trying to complete an impossible task.
It's basically a precursor variant of the cosmological argument (Aquinas or Kalam) against the impossibility of an infinite regress.
And yes, there are modern mathematical arguments the conclusions of which are that an infinite number of tasks can be completed in a finite amount of time (a nod to your own insight concerning Zeno's failure to incorporate the temporal); but that is as may be.
A lovely piece, as usual.
That said, I've always understood Zeno's paradox in what you might call the opposite way: as an argument NOT that Achilles will never reach his goal, but only approach it asymptotically; but rather, that Achilles cannot even begin moving.
It's basically a reductio ad absurdum of the assumption that space is comprised of an actually infinite number of discreet points.
Zeno's saying, look, if that were true, then in order for Achilles (or anyone, for that matter) to get anywhere - i.e. any arbitrarily distant point from his starting point - he would first have to get halfway there. but before he could get halfway there, he would first have to get halfway to the halfway point. and so on, ad infinitum. and because an infinite number cannot ever be traversed by incremental addition, Achilles would be stuck at the starting line of the race, trying to complete an impossible task.
It's basically a precursor variant of the cosmological argument (Aquinas or Kalam) against the impossibility of an infinite regress.
And yes, there are modern mathematical arguments the conclusions of which are that an infinite number of tasks can be completed in a finite amount of time (a nod to your own insight concerning Zeno's failure to incorporate the temporal); but that is as may be.
I just thought it was worth a comment :)