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If Zeno seemed to prove our perceptions cannot be trusted, how, then, can/does an Empiricist justify faith in their perceptions? I'm looking for various solutions (or justifications in the face of the paradoxes to maintain Empiricism) of these issues.

NationWidePants
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    Which paradoxes? There are many and they cannot so far as I know be reduced to a single case. –  Aug 10 '16 at 22:58

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Not everyone agrees that the paradoxes have been solved, but they aren't strictly empirical problems either.

There's a sizable school of thought, myself mostly included, holding that calculus basically solves the problems. Most of Zeno's paradoxes are founded in the claim that one cannot completely infinitely many steps in any finite time - but calculus, and convergence of limits, especially as refined by Weierstrass, tells us that an infinite totality of steps may be traversed in finite time given their convergence in the limit. Thus the puzzle becomes a prompt for calculus to come in, and that's that.

From a different direction, note that the paradoxes depend on an infinitely fine-grained world. While quantum mechanics still has a way to go on figuring this out (and may be in practice impeded from ever doing so), it's arguable that the universe operates with a particular resolution. If this is the case (and as far as we know, it might be) then the intial premise of Zeno's paradoxes - that you can cut the universe up as many times as you want in any dimension - falls apart. In the same vein, it's worth noting that Aristotle claimed the ancient Atomists were inspired by Zeno's paradoxes to posit the existence of such limited resolution (On Generation and Corruption 316b34).

Last, I'd like to suggest a borderline metaphilosophical perspective: that Zeno's paradoxes are "merely academic". Obviously, whatever Zeno might say, if I jump off a cliff I will in practice hit the ground hard. An empiricist could take this reasoning and say, "look, your little paradoxes are fun to ponder, but watch this: I'm walking. I'm traversing infinitely many points in finite time. Whatever the paradoxes may say, time and time again my perceptions are proving reliable (within reasonable bounds). We can hardly permit our scientific enterprise to be held back by a thought experiment totally divorced from reality!"

This last thought reminds me of an apocryphal tale I vaguely recall: a fellow ancient, first confronted with Zeno's paradoxes, offers an incredulous stare. They take a step forward, look about significantly, and declare, "disproved!"

commando
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    Always curious to know how the infinite divisibility of the mathematical abstraction of the real numbers resolves Zeno, since no physical theory proves the infinite divisibility of the physical world. – user4894 Aug 10 '16 at 19:53
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    @user4894 suppose Zeno is sound. Then the physical world is infinitely divisible, since Zeno depends on that. Then, the abstraction accurately models reality and resolves the paradox. Suppose conversely that Zeno is unsound. Then we're done, no paradox to resolve. – commando Aug 10 '16 at 20:01
  • @commando: not so sure that the world is as infinitely divisible as you suggest - what about atoms and quanta? Surely they suggest that there is some limit to division? – Mozibur Ullah Aug 10 '16 at 20:08
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    @MoziburUllah, see the second paragraph? –  Aug 10 '16 at 20:10
  • @commando Cantor and Weierstrass both disown infinitesimals in different ways. Cantor did not like the idea that multiplying a transfinite number by an infinitesimal was undefined, and he wanted the former to be integrated into math first. Weierstrass proved you never really need them, by changing infnintesimals to implied limits. Do you mean Cauchy and (then, much, much later) Robinson? –  Aug 10 '16 at 20:12
  • @jobermark perhaps my casual use of "infinitesimal" was sloppy. I mentioned them particularly because Cauchy's attempts at formalizing calculus were incomplete, specifically w.r.t. continuity vs. uniform continuity. Weierstrass cleaned that up, which is why I mentioned him. – commando Aug 10 '16 at 20:18
  • @commando: that's an old solution, see my answer - it's what I refer to as the 'method of exhaustion'; it's this answer - or one line of thought on what infinitesimals are - that was tightened up by Cauchy & Weierstrass; dunno what uniform continuity has to do with it, it seems irrelevant - can you explain? – Mozibur Ullah Aug 10 '16 at 20:23
  • @MoziburUllah it probably is irrelevant - it's been a while since my last analysis class. I'll clean up my own answer. – commando Aug 10 '16 at 20:44
  • @commando: it just goes to show that it's not only continental philosophy that can be accused of a mis-use of concepts and obfuscation; it can happen even in the scientific disciplines, no? I take that, though, as a given. – Mozibur Ullah Aug 10 '16 at 20:48
  • @MoziburUllah oh, of course. Analytic philosophy is quite vulnerable to misuse of concepts by falling on the low end of the Dunning-Kruger effect in a wide range of sciences. I don't think anybody here was denying that, – commando Aug 10 '16 at 20:51
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    Calculus doesn't solve the problem--or at least, doesn't solve the general problem. The issue is that you can think of the walk as a set of "staccato" steps with an infinitesimal pause between them. Then you can't get a nice continuous function any more. See the SEP on Supertasks and look for Grunbaum's 1969 paper. http://plato.stanford.edu/entries/spacetime-supertasks/ –  Aug 11 '16 at 16:04
  • @shane I would agree that calculus does not seem to solve the issue. – NationWidePants Aug 15 '16 at 13:53
  • @commando there is one other problem, you don't seem to have proposed. you're last premise seems to suggest that because we see a body hit the ground or observe a person walking an infinite number of points in a finite time that the paradox breaks down, but what if Zeno wasn't speaking of issues in reality, as this would perhaps be absurd, but a flaw in the observer of said reality, an untenable position we all suffer from. – NationWidePants Mar 02 '17 at 11:03
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Aristotle can be considered as one of the earliest Empiricists, or perhaps better, that empricism was one side of his many-sided philosophical persona.

Aristotle already points out that the method of exhaustion is an adequate solution; but he considers that this isn't a sufficient resolution when all aspects of the situation are properly considered; he suggests that a solution lies in a proper consideration of what determination & indetermination means; and this seems to have a family resemblence to Whiteheads 'occasions of determination' particularly in the analysis of motion.

Mozibur Ullah
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