On 08/30/2003 06:27 PM, Chuck Britton wrote:
>
> Mathematicians feel the need to cover
> infinite series arguments in great detail in
> order to justify the calculus that physics
> respects so much.
Maybe so ... but bringing up Zeno in this
context is a giant leap backwards.
Summing the series to find the outcome of
the race is like using a machete to swat
flies. I suppose it could be done, but
it's not the optimal way to do it, and Zeno
applied the non-optimal procedure wrongly
anyway.
More generally: never use a weak argument
to support a strong one.
> Convincing the general populace (and
> perhaps a few dumb jocks) that the infinite and
> the infinitesimal are important elements of the
> 'real world' is not an easy task.
It is not made any easier by bringing Zeno into
the discussion.
But I agree that it's not easy. Offhand I can think
of only very few examples of arguments involving
taking the limit that are interesting and nontrivial,
yet understandable by beginners. [Trivial refers to
cases where the limit of f(x) (as x goes to k) is
just equal to f(k)].
The two examples that come to mind are
-- (1+x)^(1/x) [plus tons of ramifications]
-- the ratio of successive terms of the Fibonacci
sequence [plus a few cute ramifications]
-- sin(x)/x (for students who already know what sin()
looks like; not all do at the point where
they first hear about limits.)
> ... presented so clearly by Zeno.
Nothing was presented clearly by Zeno. He was just
plain wrong.
We must all appreciate the distinction
-- I know an awful algorithm for calculating X, versus
-- X cannot occur.
Zeno got this wrong.
None of this has anything to do with Banach-Tarski.