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What is Aristotle's refutation/objection/solution to Zeno's paradoxes?

The following is all I could find in around in the Internet:

Why is Aristotle's objection not considered a resolution to Zeno's paradox?

"Aristotle's solution was largely accepted until the end of 19th century when Cantor and Dedekind formalized the notion of continuum in terms of set theory...Thinking back, one realizes that Aristotle's solution was always incomplete...".

http://physics.bgsu.edu/~gcd/Spacetime3.html

"Unfortunately, Zeno's work has been lost. All we possess are paraphrases by commentators and critics. The paradoxes of motion are known only from their formulation by Aristotle, whose purpose was to criticize and refute them. This presents a number of problems of interpretation, not the least of which are that (1) we do not know against whom the paradoxes were directed, and (2) we do not know exactly how Zeno originally formulated the paradoxes."

The offered bibliographic reference therein is unfortunately unclear.

The following seems to distill better Aristotle's argument, though it seems to:

https://ancientphilosophy.wordpress.com/2011/04/20/aristotles-solution-to-zenos-paradox-of-the-runner/

In my opinion, it is not clear enough, and it offers an accessible reference to only half of the presentation: Barnes, J. (1979) The Presocratic Philosophers, London: Routledge. p. 262.

  • I am not sure what you are looking for can be found on a Q&A site, we can not outdo professional scholarship and encyclopedias. Here is IEP commentary on Aristotle's solution. – Conifold Feb 24 '18 at 0:31
  • Related but somewhat different: philosophy.stackexchange.com/questions/30154/… – user4894 Feb 24 '18 at 2:05
  • Paradoxes cannot have simple, clear cut solutions... otherwise they are not paradoxes at all, but simple mistakes. – Mauro ALLEGRANZA Feb 24 '18 at 9:35
  • @MauroALLEGRANZA - I'd agree. In Zeno;s case the mistake may be our usual view of motion, which seems to have been his point. – PeterJ Feb 24 '18 at 11:45
  • @MauroALLEGRANZA A paradox need only seem contradictory or absurd. Zeno's paradox obviously does not show anything other than his particular model of reality did not suffice; this does not rule out a simple solution from someone with a better model. – Veedrac Feb 24 '18 at 14:20
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As explained in IEP's entry regarding Zeno's Paradox, current solution (aka Standard Solution) is based on the mathematics of the infinite, developed after 17th Century.

Current mathematical solution makes sense of an infinite sum having a finite amount.

This is not so for ancient mathematics and philosophy, as well as for Aristotle: either the quantities that we have to add are zero, in which case the total of the infinite sun is zero, or the quantities are not zero, in which case the sum of an infinite of them must be infinite.

Aristotle's solution is based on the dichotomy between actual and potential infinite:

For motion…, although what is continuous contains an infinite number of halves, they are not actual but potential halves. (Physics, 263a25-27). …Therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible: if they are potential, it is possible. (Physics. 263b2-5).

According to A, the finite distance is only potentially infinite, in the sense that we can - in thought - imagine to divide it an infinite number of time.

But it is actually finite, and thus any mover, moving with a finite speed, can traverse it in a finite amount of time.

  • As a person who knows enough Calculus to resolve the paradox myself, the reason why I wanted to learn what kind of argument Aristotle's came up with 2k years before Cantor, Riemann, Newton and Leibniz is that I was expecting something really clever (if understandably primitive). Now that I know, until I find some value in that "potentially infinite" idea, I'm disappointed. – Katerl3s Feb 25 '18 at 3:52
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    @Katerl3s - Aristotle's argment is what it is... and, in general, Aristotle is quite clever (see A's Logic). The idea of potential infinite is quite clear and very natural: see the unlimited possibility of iterating the baisc operation of adding one number. See e.g. the post is-there-a-formal-distinction-between-potential-and-actual-infinities. – Mauro ALLEGRANZA Feb 25 '18 at 8:05
  • @Kater3s: Heisenberg was actually very impressed with Aristotles argument. So perhaps its not quite as primitive as you might think. Also, the method of exhaustion which is the basis of calculus was understood by Aristotle as he mentions it in passing as an adequate answer to Zenos paradox (but not sufficient). – Mozibur Ullah Feb 26 '18 at 19:01

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