Re: OT: White Goddess

On Thu, 12 Apr 2001, Andreas Johansson wrote:

> bjm10 wrote:
> >On Wed, 11 Apr 2001, Yoon Ha Lee wrote:
> >
> > > On Wed, 11 Apr 2001, Andreas Johansson wrote:
> > >
> > > Suppose you have a regular polygon with n sides.  (I think you could get
> > > by with a weaker condition but this will suffice.)  The "limit" of the
> > > polygon as n goes to infinity is a circle.
> >
> >But it isn't a polygon with an infinite number of sides, because the
> >length of each of those sides would have, perforce, to be zero, which
> >means that the circumference, being a sum of zeros, would be zero.
>
> This makes no sense, does it? If a polygon with a given circumference have
> an infinite number of sides, it follows that the length of each side is
> infinitely small (the length'd be "infinitesmal").
Oh dear...my message on precisely this was being typed up when this
message came, ergo I didn't read yours first and I was redundant.  Ah well.

> According to your reasoning, if I took a line 1m long and divided it into an
> infinite amount of pieces the total length of the fragments would be zero.
> 1m has disappeared without anyone removing any length, eh?
Zeno!  <G>

> >Then say "transfinite", in that case.  "Infinity" is not a number.  It is
> >not a quantity.  It should not be treated as if it were.
>
> Well, infinity certainly have cardinality (ok, I know that different
> infinities can have different cardinality but lets keep things simple now
> shall we?). You can count with it much like you can with any (other) number.
> Infinite quantities aren't hard to find (the number of natural numbers comes
> to mind).
<nod>  Yes--I find infinities fun to think about.  :-)  The "rules of
arithmetic," for example, are quite different with infinite sets.  (The
even numbers are in 1-1 correspondence with the natural numbers are in
1-1 correspondence with the odd numbers are in 1-1 correspondence with
integral multiples of 3, etc.)

As I said in my last message, how you treat infinity really depends on
what mathematical context it appears in....

YHL