You may have noticed that in previous sub-units
I've been talking a lot about complexity
in an informal way.
But I haven't yet defined the term "complexity".
That's for a reason. It turns out that
"complexity" is hard to define.
Or, more precisely,
it has too many different definitions
in different fields.
So how do complex systems researchers
measure the complexity of a system?
Seth Lloyd's paper called
"Measures of Complexity: A Nonexhaustive List"
gives something like 42 different definitions
or ways of measuring complexity,
including Shannon information,
algorithmic complexity, fractal dimension,
thermodynamic depth, etc., etc.
Is there a single comprehensive and useful
definition of complexity?
It's very doubtful.
These different definitions are useful
for measuring different aspects of systems.
In this course, we'll talk in depth
about two of these:
Shannon information and fractal dimension;
how they're used, and why they might be useful.
We'll also discuss the general problem
of defining and measuring complexity
in the real world.
Before I go on,
I want to mention the ideas about complexity
in one excellent, classic, and very prescient paper,
called Science and Complexity,
written by the American mathematician
Warren Weaver in 1948.
Weaver divided the problems of interest in science
into three categories.
The first category he called Problems of Simplicity.
These are problems that involve
just a few variables.
Some examples might be
relating pressure and temperature
in thermodynamics; or, in electricity,
relating current, resistance, and voltage;
in population dynamics,
relating population versus time.
These are all problems that were dealt with
in the 19th and early 20th centuries,
in physics, chemistry, biology, and so on.
Then Weaver goes on to a second category
that he calls Problems of Disorganized Complexity.
These are problems involving billions
or trillions of variables.
So one example would be understanding
the laws of temperature and pressure,
as arising from trillions of disorganized
air molecules in a room or in the atmosphere.
These are understood through
taking averages over the large set of variables.
When we look at understanding temperature,
we don't look at the particular position
and energy of every individual air molecule.
Rather, we understand temperature as
the average energy over the trillions of molecules.
And the science of averages comes under
the rubric of statistical mechanics,
which deals with these kinds of problems.
The key here is that we're assuming
very little interaction among the variables.
That's what allows us to take meaningful averages.
In the case of the temperature of a gas,
the whole is the sum, or equivalently,
the average of the parts.
Weaver's last category was the problems
of organized complexity.
These are the problems that include
the examples I gave earlier,
the problems of interest
to complex systems researchers.
These are problems that involve
a moderate to large number of variables.
But the key here is that,
due to their strong nonlinear interactions,
the variables cannot be meaningfully averaged.
Again, we'll talk more precisely about
what "nonlinear" means in the next unit.
Weaver characterized these as
"problems which involve dealing simultaneously
with a sizable number of factors
which are interrelated into an organic whole".
So this really gets at the notion of emergence.
This "organic whole" refers to
the emergent behavior of the system.
In his paper, Weaver gives a beautiful list
of questions as examples of problems
of organized complexity.
It's very striking to note that,
even though Weaver's paper
was published in 1948,
all of these problems point to issues
that are still open questions
in complex systems science
almost seven decades later.
I'll just run through a few
of his questions here.
What makes an evening primrose
open when it does?
What is the description of aging
in biochemical terms?
What is a gene,
and how does the original genetic constitution
of a living organism
express itself in the developed characteristics
of the adult?
On what does the price of wheat depend?
How can currency be wisely
and effectively stabilized?
How can one explain the behavior pattern
of an organized group of persons
such as a labor union,
or a group of manufacturers,
or a racial minority?
Weaver went on to say
"These problems... are just too complicated
to yield to the old 19th century techniques,
which were so dramatically successful on
two-, three-, or four-variable problems
of simplicity.
These new problems, moreover,
cannot be handled
with the statistical techniques so effective
in describing average behavior
in problems of disorganized complexity."
And going even further, Weaver said,
"These new problems -
and the future of the world
depends on many of them -
require science to make a third great advance,
an advance that must be even greater
than the 19th-century conquest
of problems of simplicity
or the 20th-century victory over problems
of disorganized complexity.
Science must, over the next 50 years,
learn to deal with these problems
of organized complexity."
Well, it's been almost 70 years
since Weaver wrote this paper.
And a major purpose of this course
is to let you know how far we've gotten
in dealing with problems of organized complexity,
and what new tools
has the science of complexity developed
to deal with them.
Even though we're not going to go deeply
into formal definitions in this class,
let's do a bit more to investigate the question,
"What is a complex system?"
which has many different possible answers.
Let's go straight to the experts.
The next sub-unit gives a sampling
of answers to this from several
of the best-known experts in the field.
Note that, while there's a great deal
of variation in the answers,
there is some convergence, as well,
on a definition.