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# [Phys-L] Re: Zeno's Paradoxes

• From: John Denker <jsd@AV8N.COM>
• Date: Mon, 28 Mar 2005 20:24:40 -0500

Ken Caviness wrote:

Of course, from calculus we now know that the sum of an
infinite series can quite possibly be finite,

That is the right answer. That is the crux of the matter.

Let me elaborate just a little bit.

We must deal with two different ideas:
a) what happens?
b) how do we know and/or calculate what happens?

It is important to distinguish between the _happening_ and the
_knowing_.

Science often involves building models. Some models are more
faithful than others.

In particular, in these Zeno problems, there are two notions of
time:
a) how long does it take for the tortoise (or arrow) to reach
its goal?
b) how long would it take you to sum an infinite series term
by term?

These are two very different notions. If we write
t = 1 + 1/2 + 1/4 + 1/8 ...
the value of t is quite finite, whereas the time it would take
you to calculate t term-by-term is quite large. The fact
that t has dimensions of time, while the running-time of your
summation algorithm also has dimensions of time, is a source of
confusion, but the confusion is easily dispelled: these are two
entirely different times.
a) the sum of the series is a faithful model of the physics
b) the time spent summing the series term-by-term is not.

We know from everyday experience that you can build an analog
computer (using a real arrow or real tortoise if necessary) that
will sum this particular series in bounded time, no problem.

There are also formal, analytic methods for summing this series.

So another philosophical point is that the existence of a bad
algorithm does not preclude the existence of better algorithms.

This is related to the paradox of Achilles and the tortoise:

Yup.

The arrow paradox doesn't translate so conveniently into an infinite series,

Actually I think it does. It looks exactly the same to me.
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