Re: sin x / x tends to 1...

On 5 Sep 2005 10:49:07 -0400, lrudolph@xxxxxxxxx (Lee Rudolph) wrote:

>David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> (not addressing me) writes:
>
>>So answer me a question. Exactly how would someone
>>with no hidden agenda re sin(x)/x _define_ the
>>length of a curve?
>
>Just to beat a dead horse a little more, here is another--and
>I hope clearer, but in any case more ample--version of what
>I would say.
>
>"Length" ought to be a (partially defined) functional on the set
>of all curves, preferably a continuous one after we have agreed on
>an appropriate topology for that set (which includes agreeing on
>what the set is!), that has a bunch of properties that I enumerated
>before. (1) It ought to agree with the given definition of length
>on a straight line segment. (2) It ought to be preserved by
>isometries and scale linearly by homotheties. (3) It ought to
>be additive (in an appropriate sense). ... And (maybe) so on.
>Such a laundry list of desiderata doesn't *define* anything,
>but if we're lucky it *characterizes* something; and, in fact,
>that list (maybe with "and so on" beefed up a bit...I'm out
>of practice on this kind of thing, as may have been obvious)
>*does* characterize "length" for rectifiable curves in R^n,
>in the sense that there is one and only one functional
>with the given property that is "what it should be" on
>piecewise-linear curves ("broken lines") and is continuous
>on the completion of the piecewise-linear curves in the
>appropriate topology.

Yes, that seems exactly right to me, although I'd insert
something about how it turns out that the relevant topology
under which the thing should be continuous is a little
trickier than one might guess at first.

>To actually have any hope of *calculating* (even theoretically),
>one of course needs a particular definition of this functional,
>and the standard one--via inscribed broken lines--of course works
>(precisely because of all the hard work that I shovelled under
>the rug in the last five lines or so of the preceding paragraph).
>But so would others, for instance, ones coming from integral
>geometry (at least, I think they would) or the distinct-but-
>similar definition that starts from a *smooth parametrization*
>of a curve (and not the geometric object, as I was implicitly
>doing above; also, not limited to piecewise-linear curves),
>integrates the length of the tangent vector on such a thing
>to get the length of such a thing, and then completes *that*
>space in the appropriate topology.
>
>The point I think I'm trying to make is what I understand to be
>the point Herman Rubin often makes, particularly about the
>integers, but also the real numbers: a characterization is
>better (because more conceptual) than a definition.

Yes, positively a characterization is better. This is exactly
why for the first 2/3 of this thread I kept saying I didn't
want to use the definition! We didn't need any definitions
when we were just talking about areas, all we needed was
the area of a polygon and an "obvious" monotonicity wrt
subsets.

And what still bugs me is that I don't see how to show
that the circumference of the unit circle is twice
the area _without_ using the definition, because I
don't see any "obvious" reason there should be any
inequality between the circumference and the length
of a circumscribed polygon (proving it may or may
not be hard, but the proof starts with the definition...)

>None of this means that *other* people in this thread don't have
>hidden agendas. But my agenda is out on the table. (And I see
>a fat chance that it's going to be generally adopted.)
>
>Lee Rudolph


************************

David C. Ullrich
.