Re: OT: White Goddess

bjm10 wrote:
>On Wed, 11 Apr 2001, Yoon Ha Lee wrote:
>
> > On Wed, 11 Apr 2001, Andreas Johansson wrote:
> >
> > > Eh, seeing that I supposed to be good at math, I should probably know
>this
> > > myself, but what's the difference between a circle and a regular
>polygon
> > > with an infinite number of sides? I definitely recall being told by my
>math
> > > teacher tellin' me they're the same ...
> >
> > 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").

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?

>As
>you say, the number of sides *approaches* infinity, but it is an
>asymptotic limit.
That's kind of the point, isn't it? An asymptot reaches its limit at
infinity.

>  A curve is a curve, not a polygon, but a curve can be
>approximated by a polygon, if one actually wants to do something
>practical.  Approximation is not identity.
I don't think anyone has argued that.
>
> > I am not certain what mathematical meaning, if any, "transcended
> > infinity" has, though.
>
>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).

                                                       Andreas
_________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

Replies