[Phys-l] singular limits (was: no units)

[John Denker <jsd@av8n.com>]

On 11/06/2009 09:26 AM, Bernard Cleyet wrote:

> I'm in the middle of jury duty, but I would recommend dropping the
> term "newtons." Zero In this case does not require units: 0 N = 0 lb =
> 0 dynes.


0 newtons is also equal to 0 furlongs.

This applies to vector character as well as to
dimensions and units:

           0 newtons in the Xhat direction 
         = 0 newtons in the Yhat direction
         = 0 cubits in the Zhat direction

HOWEVER ... this is an example of a singular limit.

Consider the predicate Q as a function of m:

  Q(m): Is m newtons different from m furlongs?

We have:
  Q(m) is true for all nonzero m.
  Q(m) is true in the limit as m goes to zero.
  Q(m) is false when m is exactly zero.

You should always be leery of anything that is true in
the limit but not true right at the limit point.  This
is called a singular limit.

In particular, physics is one of the natural sciences,
which means it is not an exact science in the sense that
arithmetic is exact.  So even though Q(m) is false when m 
is exactly zero, it is true when m is _approximately_ zero.

Singular limits are a problem.  Usually they are just
problems with the terminology, but sometimes there are
genuine singular limits in the physics.  A famous example
concerns flying through a viscous medium:  a wing works
better and better as the viscosity gets smaller, but if
the viscosity were strictly zero it wouldn't work at all.

This example, and the infamous "worm in the apple" example
are nicely discussed by Berry (yes, THAT Berry) at
  http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry341.pdf

Highly recommended.