at least tangentially P-related: Class of all classes

[Monte Davis]

David Payne writes:

> > the "curve which is everywhere continuous and nowhere 
> > differentiable"... Nice ring to it.
> 
> That would be the Weierstrass function, which is "a 
> pathological example of a real-valued function on the real 
> line. The function has the property that it is continuous 
> everywhere but differentiable nowhere" 
> (http://en.wikipedia.org/wiki/Weierstrass_function).

Not differentiable = crudely, not amenable to that basic operation of
calculus, deriving a graph line's slope (ratio of "rise":"run") over smaller
and smaller intervals. The most obvious bad actors would be discontinuous --
dashed or intermittent lines, which have no Y value for a given X because
the function is doing something illegal like dividing by 0.

Naturally, over-generalizing like the rest of us, mathematicians had grown
to think "continuous" and "differentiable" must go together when Weierstrass
rubbed their noses in counterexamples: functions yielding graph lines that
are *all* spikes, *all* cusps where the slope has no meaning. no matter how
small the interval. And of course, once the bad juju of the "pathological"
label wore off, people started finding such functions everywhere.

>  I am pleased to learn that 
> "pathological" has a mathematical definition: "In 
> mathematics, a pathological phenomenon is one whose 
> properties are considered atypically bad or counterintuitive" 
> (http://en.wikipedia.org/wiki/Pathological_%28mathematics%29).

Likewise, Benoit Mandelbrot dwelled on some "pathological" geometric forms
(Koch snowflake, Sierpinski sponge, Peano dusts etc.) until it turned out
they were descriptive of lots of everyday phenomena. Likewise, a century of
dynamical-systems work from Poincare and Liouville through Abraham, Smale &
co. turned "chaotic" systems from rare curosities into the stuff of
turbulence, weather, solar systems over hundred-million-year spans, etc.

In retrospect, it looks like the drunk searching for his keys under the
streetlight not because he lost them there, but because that's all he can
do. Yet the circle of light *does* expand.