Re: OT: White Goddess

On Thu, 12 Apr 2001, Bryan John Maloney wrote:

> On Wed, 11 Apr 2001, Yoon Ha Lee 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.  As
> you say, the number of sides *approaches* infinity, but it is an
> asymptotic limit.  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.
Well, you *could* go the nonstandard analysis route.  The length of each
side would be infinitely small.  <sigh>  This is where Newton & Leibniz &
co. got bogged down with the foundations of the calculus (which, I might
add, in its original conception *did* deal with infinitesimals, as they
were called).  Your argument is reminiscent of one of Zeno's paradoxes,
which isn't a paradox after all when the mathematics are more clearly
understood (and this took any number of centuries, and who could blame us
poor humans?).  If I have time later I can go into detail, or if someone
else has the argument in memory (I would have to refresh mine) as to why
Zeno was wrong (and why the above argument doesn't quite work), feel free.

Er...if you want a source, _The Mathematical Experience_ by Reuben Hersh
and Philip? Davis goes over this (as well as an introduction to
nonstandard analysis, which does deal with such infinities and
infinitesiamsl) pretty well and not too technically.

> > 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.
Um--infinity isn't a real number, or a natural number, or a complex
number.  Nevertheless there *are* number systems in which various
infinities are treated mathemtically as numbers.  (Thank you, Cantor,
even though the poor man died in an asylum, or at least spent a lot of
time there.)  If infinities are treated as numbers in a "smaller" system
you can have screwy things happen.  (I might mention that physicists
working in general relativity regularly use "renormalization," which is
dividing infinity by infinity.  Mathematically it's screwy and there's no
foundation for it--yet?--but physicists don't care about mathematical
niceties--witness the horrible "bra" and "ket" Dirac notation
<shudder>--and for what it's worth, renormalization *works* in that it
produces answers that are correct so far as we can tell.)  You can have
aleph-nought and Mahlo ineffable cardinals and god knows what else; these
things *are* called transfinite numbers and are treated, I suppose, as a
"special kind" of number by mathematicians.  OTOH, we are also the people
who abuse words like "normal," "regular," "group," "ring" and "field," so
what can I say?

But a mathematician *would* say infinity is a number, in the appropriate
system.

YHL, less than 2 mos. away from a bachelor's in math
and I'm sure there are other mathematically-trained people on this list
who can kick my tail on the subject, too

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