Editing Zeno's paradoxes (section)

=== In classical antiquity ===
According to [[Simplicius of Cilicia|Simplicius]], [[Diogenes the Cynic]] said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions.<ref name=":2" /><ref name=":1" /> To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Throughout history several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

[[Aristotle]] (384 BC–322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.<ref>Aristotle. Physics 6.9
</ref>{{failed verification|reason=In the section cited, Aristotle says nothing about the distance decreasing |date=October 2019}}<ref>
Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a [[Harmonic series (mathematics)|harmonic series]], while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a [[divergent series]], the sum of which has no limit. {{Original research inline|date=October 2020}} Archimedes developed a more explicitly mathematical approach than Aristotle.</ref>
Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").<ref>Aristotle. Physics 6.9; 6.2, 233a21-31</ref>
Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."<ref>{{cite book |author=Aristotle |title=Physics |url=http://classics.mit.edu/Aristotle/physics.6.vi.html |volume=VI |at=Part 9 verse: 239b5 |isbn=0-585-09205-2 |access-date=2008-08-11 |archive-date=2008-05-15 |archive-url=https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html |url-status=live }}</ref> [[Thomas Aquinas]], commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."<ref>Aquinas. Commentary on Aristotle's Physics, Book 6.861</ref><ref>{{Cite book |last=Kiritsis |first=Paul |title=A Critical Investigation into Precognitive Dreams |date=2020-04-01 |publisher=Cambridge Scholars Publishing |isbn=978-1527546332 |edition=1 |pages=19 |language=en}}</ref><ref>{{Cite web |last=Aquinas |first=Thomas |author-link=Thomas Aquinas |title=Commentary on Aristotle's Physics |url=https://aquinas.cc/la/en/~Phys.Bk6.L11 |access-date=2024-03-25 |website=aquinas.cc}}</ref>