🚀 go-pugleaf

In message <32F1965C.382A@worldnet.att.net> Larks wrote:

> Xcott Craver wrote:
> > 
> > In article <5cmpbo$nds@agate.berkeley.edu>,
> > Louis Hom <lhom@nature.berkeley.edu> wrote:
> > >So, I'm trying to collect posters' favorite Things in Math & Science.
> > >       For example, one of my favorite things is the solid that you get
> > >when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
> > >infinite surface.  So if you had a container that shape, you could fill it
> > >up with some finite quantity of paint, but it would never be enough to
> > >actually paint the surface of the thing.
> > 
> >         Or, a cylindrical can whose base is the Koch snowflake would
> > do the trick, and would actually fit on a shelf.
> > 
> >         I think it would be more mind-boggling to point out that
> > the can holds a finite amount of paint, but it takes an infinite amount
> > of paint to paint THE INSIDE.  Aren't you technically painting the
> > inside when you fill it up with that finite amount of paint???
> > 
> >         Of course, this problem stems from the abstraction that the
> > layer of paint you lay down on a surface is infinitely thin, which
> > isn't true in reality, and serves as an amusing reminder that areas
> > and volumes are not comparable mathematically the way they are in
> > the real world.
> 
> Exactly.  The problem stems from extending mathematics to the physical
> world while retaining multiple assumptions which apply only in 
> the abstract mathematical sense.  If I may get philosophical for a 
> sentence or two, that's where mathematics and physics part company.  
> In physics, a result must make sense; in mathematics, it must follow
> from mathematical manipulation and, if it makes no sense, it is 
> added to the list of examples of things deserving of a Nobel prize,
> if only math had one.

But surely the interesting thing about this paint can happens without
extending into the real world.

If you fill the can with mathematical paint (which can be arbitrarily thin),
then at every point there is a finite (if arbitrarily small) depth of paint;
but there is no uniform finite thickness of paint with which you can use
that paint to paint the surface.

-- 
Mike Williams
http://www.econym.demon.co.uk/