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how metrics determine the knowability of a phenomenon is the domain of epistemology.
But most scientists are not particularly interested in philosophy; a scientist's or
engineer's concern is typically restricted to what one can record using a piece of mea-
suring equipment, such as a ruler or thermometer, or even a finger on a pulse. The
philosophical discussion is traditionally ignored, being pushed into the realm of meta-
physics, and the scientist/engineer moves on. It is only when the traditional definition
of a quantity leads to inconsistencies in experiment or theory that the working scien-
tist decides to reexamine the assumptions upon which measurements or calculations are
based. In the last half of the twentieth century Benoît Mandelbrot literally forced sci-
entists to reexamine the way they measure lengths in space and intervals in time. His
introduction of the geometric concept of the fractal provided a new tool with which
to examine the physical, social and life sciences to better understand the world. In his
monograph [
45
] Mandelbrot merged mathematical, experimental and scientific argu-
ments that, taken together, undermined the traditional world picture of the physical,
social and life sciences. For example, it had been accepted for over a century that celes-
tial mechanics and physical phenomena are described by smooth, continuous and unique
analytic functions. This belief was part of the conceptual and mathematical infrastruc-
ture of the physical sciences. The changes of physical processes in time were modeled
by webs of dynamical equations and the solutions to such equations were thought to
be continuous and differentiable at all but a finite number of points. Therefore the phe-
nomena being described by these equations were themselves also thought to have the
properties of continuity and differentiability. Thus, the solutions to the equations of
motion such as the Euler-Lagrange equations, or Hamilton's equations, were believed
to be analytic functions and to represent physical phenomena in general. Perhaps more
importantly, the same beliefs were adopted by the social and life sciences as well.
Mandelbrot presented many examples of physical, social and biological phenomena
that cannot be properly described using the traditional tenets of dynamics from physics.
The functions required to explain these complex phenomena have properties that for a
hundred years were thought to be mathematically pathological. He argued that, rather
than being pathological, these functions capture essential properties of reality and there-
fore are better descriptors of the world than are the traditional analytic functions of
theoretical physics. The fractal concept shall be an integral part of our discussions,
sometimes in the background and sometimes in the foreground, but always there. At
this point, reversing perspective, let us state that the mathematical pathology attributed
to what are now called fractals can be taken as one of a number of working definitions
of complexity in that a complex web often has fractal topology and the corresponding
measured time series have fractal statistics.
2.2.1
Fractals
We make use of a simple example of a fractal process in order to develop some insight
into the meaning of scaling relations and to develop nomenclature. There are three dis-
tinct kinds of fractals. Some are purely geometric, where the geometric form of an object
at one scale is repeated at subsequent scales. Then there are the dynamics fractals where
