Tom (and I doubt anyone else is interested):

In general, the only way 'reason' works in the direction of truth (outside pure mathematics, I suppose) is to reconcile two very different modes of thought: inductive and deductive.

Usually, a 'paradox', be it as trivial as Zeno's arrow or Zeno's runner, is indication that the set of hypothesies about nature from which you derived the paradox is untrue and/or inconsistent. Zeno's hidden assumption is that space is infinitely quantizable, divisible and discrete, which it is not- not really. (Planks constant might be dragged kicking and screaming into this).

Now if my 10-year-old knowledge of ancient philosophy holds true, Zeno was an appologist for Parmenides (sort of like the first gutless brown-nosed Grad Student of all academic history... Even pror to Plato!), and Parmenides was pushing this 'all-is-one' pseudo-spiritual metaphysics all over Greece, to much popular acclaim. A result of this tripe was that you had to consider all differentiation, motion and decay as illusion.

So along comes Zeno, who is most certainly smarter than poor quasi-mystic Parmenides, and he starts applying retoric, logic and mathematics in an attempt to prove Parmenides point for him, as Parmenides is not very well recieved by the rational folks down in Athens.

So Zeno, in order to show those durn logical Athenians who is who accepts their [the atomist] premise of divisibility and attempts to show all of them, logically speaking, that if you accept these premises that fallacy follows. And he does a really fantastic job. As simple as the four main contradictions are (Dichotomy, Achilles, Arrow, Stadium), they are devistating in their result.

Yes, obviously, they are FALSE, of course, but they are FALSE and logically consistent and seem to start from evident premises that conform to best observation. So what are we to do? Accept that there is no divisibility or accept that there is no motion or change? (neither... we eventually accept that what we mean by motion is more sophisticated than we see with our own two eyes).

In order to really explain how you can have the scenarios of the Stadium and the Arrow WITHOUT paradox, you need a notion of a four dimensional universe and you need to have notion of relative time and relative simultneity. To get this you need a much more interesting mathematics, a BUNCH of empirical observations about the relative speed of light and a GERMANIC (not greek) ugly loonie with wild hair (all the greeks had back then were Euclid and few ugly greek loonies with wild hair)

also, Planks constant probably helps...

I think one of the four, Achilles I think, can be resolved using caluculus.

A lot of very fundamental Math and Logic (Calculus, for one) came into being in the next thousand years while people dealt with these obvious, but still consistent and sound paradoxes.

But the point is the power of Zeno was not his ability to use logic to argue obvious fallacy, but in his use of logic to lay out foundational problems with our understanding of the universe that MUST be addressed. (sort of like the Bronze Age Hilbert)

And the point of addressing paradox formed by logic is that empirical observation and refinements of reference (as in viewpoint), over time, can refine or shift the axioms from which you reason.

And also, the articulatory power (esp. lack of power) in the language which you use to reason (Logic/Math) can skew the conclusions you come to (and we'll sweep Goedel and Turing off to the side for now so they don't wreck our tea party entirely).

The end result is that progress to truth is measured in dead or resolved paradoxes, In changed reference points, in more elegantly stated axioms/hypothesies about the universe, and in the boundaries you discover (Goedel and Turing again) about what reason is at all good for.

In short, the moral of the (his)story is that only through induction, deduction and reflection on inductive and deductive method can we really, as a species, tell our ass from a hole in the ground.

Many people want to write up a bill of sale that says Zenos paradoxes are proof entirely that them-thar damn Mathematicians, Logicians and Philosophers are a-jus' full of sheeit cause dey say dat a man can't outrun a damn turtle. But thats not the point. Zeno was the Russel of his day- Worse, we ironed out the implications of Russels paradox rather quickly, it took over 1000 years to really put Zeno to bed.

The point is that common sense/fundamental observation tells us a lot of things that either are blatantly wrong, or lead to blatantly wrong conclusions when taken as axioms of a formal language. Common sense once said the world was flat, that the sun orbited the earth, blacks were not human and all sorts of nonsense. Common Sense has lead to all sorts of moral and physical tragedies for mankind.

While I must use 'common sense' all the time in the universe, every moment of every day, I try to mantra myself into understanding how little I grok about the universe, and how inadequate common sense is to apprehend reality as it is. Meanwhile, I have to trust it.

Thi is what happens when you give an ex-philosopher a bit of scotch and let him log on to habitablezone, BTW. I bet people get all in a tizzy about how uncool/wrong/etc this post is... But what can I say, You just had to drag Zeno in...

Davin L.