CHAOS AND CRIME

LECTURE 002

CHALLENGES
for Postmodern Criminology

Dec., 1996

INTRODUCTION:       In this Chapter, we will explicate a set of
terms with which to grasp the basic ideas of chaos theory as they
help enrich our understanding of the complexity of crime.  Buried
in these new ideas about how nature and society work are a great
many challenges to the genius and ingenuity of the next generation
of criminologists.  With the tools of chaos data analysis, we may
well be able to sort out the changing sources of crime, distinguish
between prosocial and anti-social innovation, offer guidelines for
social policy and, thus, augment human agency to some limited
extent.  All these possibilities play off of modernist approaches
to the knowledge process which promise quicker and more tidy
answers to questions of causality but which freeze such correlates
into a theoretical model unalterable by time and social change.

     A postmodern word of caution in thinking about these and other
terms in science and the arts; whenever the incredibly complex data
of everyday life are compacted into words, information is lost. 
Still more information is lost when disciplines tear the rich and
intricate fabric of life into narrow strips and confine one sector
of life into its own conceptual prison.  Summary statements, formal
propositions, parsimonious theory and algebraic formulations
further isolate and impoverish the knowledge process.  Modern
science buys its elegant truisms at the expense of the complexity
of nature and society; unless it turns its own concepts back upon
its own intellectual product, postmodern science can make the same
epistemological error. 

     Postmodern phenomenology teaches that, whenever one set of
concepts are used; another set of concepts is foregone; whenever a
given way of seeing is adopted; all other ways of seeing...and thus
knowing, recede into the background.  Out of the incredibly rich
and interconnected social life world fitted inside a much larger
geo-ecological system, there is always a politics in choosing and
discarding of terms; in selecting and ignoring of variables; in
construction and refining of methods.  Thus, I do not want to claim
that the knowledge process in criminology must center only these
terms set forth here; I do want to say that these terms and the
ideas which use them offer new and valuable intellectual leverage
with which we may prise open the data we collect and find things
that could not be found using other, more familiar scientific terms
and methods.  

     If there is any one truth in the postmodern philosophy of
science which is aborning, it is that there are any number of valid
ways to rotate the complex realities before us; any number of ways
to slice such complex realities into useful concepts; any number of
pathways to generate data and human understanding and, as well, any
number of useful ways to talk about the complexity of nature and
society.  For postmodern sensibility, poetry lies at the near end
of the knowledge spectrum at which theory is found while policy and
practice lay at the far end.

THE FIRST CHALLENGE.  The first challenge to criminologists in the
21st Century is to discard modernist notions of science and knowledge
in favor of a postmodern philosophy of science which honors change,
diversity, difference and uncertainty.

Modern science, since Newton, has sought for universal covering laws
which embody the quest for absolute certainty...absolute Truth as
Hegel put it.

The real world is comprised of dynamical systems which are far more
complex, far less stable, far too surprizing to provide those absolute
covering laws so dear to the heart of those who would know and would
control all nature and all society.  That is just not on.  The new
philosophy of science before us has much more modest aims.  The new
generation of criminologists have far more difficult tasks.  The
tutorial below can help.

Tutorial     Our first concept, after a brief tutorial on Chaos theory, 
is that of an attractor.  One would do well to think of each term
below itself as an attractor, each use of which has some varying
self-similarity; differing basins of meaning; qualitative change in
meaning as one changes scale as well as having wide spaces through
which much meaning may escape.  If we turn the language of chaos
theory back upon itself, we can benefit from the insights it
provides without the arrogance and totalizing tendencies of other,
more linear and reifying language systems; other more formalistic
and totalizing theoretical systems.

Chaos Theory:  A science which deals with the complex harmonies and
     disharmonies exhibited by natural and social systems.  It is
     the study of the changing ratio of order and disorder in an
     outcome field.  Chaos theory focuses upon states with multiple
     periods or without predictable periodicity.  Chaos research
     studies the transitions between linear and nonlinear states of
     such dynamical systems. [From Chaos, <L. <G. abyss; from which
     our word, chasm also comes.  The presumed original state of
     disorder of the unformed world.  Funk and Wagnalls
     Dictionary.]

     If one had to pick one definition of this new science of
complexity, one might do well to simply say that it is the study of
the changing ratio between order and disorder in nature and
society.  It is particularly important to remember that disorder is
common to all complex systems; equally important is to keep in mind
the great order  found in those same complex and uncertain systems. 
In a torus, while there is disorder, there is mostly order.  In a
butterfly attractor, while there is some uncertainty at key regions
in the outcome field, there great certainty in most regions.  Even
in deep Chaos, beyond the fourth bifurcation point there are forms
of order only just now being discovered.

     Human beings, concerned with survival or security, tend to
look at change and see great disorder since, for most of human
existence, a small change in food supply or temperature could and
still can produce a large change in survival chances.  Hence the
word Chaos has much greater emotional impact than warranted by the
relatively small changes in the ratio between order and disorder
found in 2n, 4, 8n, or 16n attractor fields.  For that reason, most
people who now work the field prefer the word, complexity, as a
label to refer to all that has come out of this new body of
science.  We will use both since both point to a body of knowledge
with great overlap however, all in all, complexity is the better
term.

     For behavioral sciences generally, it is important to note
that there are very stable structures to be found in apparently
disordered data.  Even in a depression with 15% unemployment, 85%
of the work force are producing.  In criminology, even in a medical
profession in which thousands of doctors file tens of thousands of
fraudulent claims amounting to millions of dollars every month,
still that fraud is but a small fraction of the totality of medical
procedures fairly prescribed and fairly billed.  For most humans
for most of human history, the ratio between order and disorder is
far, far more favorable to the familiar and the known than to
surprize and mystery and in that changing ratio, order is central
to the human project.

     Having said that order is far more on the side of survival
than human perception or conception may say, I need to say also,
that nonlinear behavior has survival potential for the human
species, indeed for every living creature, that conservative
thought can grasp.  In a wide variety of situations, it is far
better to exhibit behavior with some nonlinearity than behavior
which is perfectly regular, perfectly predictable, perfectly
ordered.  The moth and the butterfly fly on their irregular course
in order to thwart predators.  Human heartbeats are chaotic in
order to more quickly respond to demand for oxygen.  T-cells
exhibit chaotic variations on protein assemblage in order to be
able to find and fit itself to the surface structure of new viruses
and bacteria.  Information itself depends upon a changing mix of
order and disorder.  And, as we descend into deep chaos, we will
find more than uncertainty; we will find creativity and even more
complexity.  

     The single most encompassing thing one might say about
nonlinearity in terms of human interest in social policy and
survival is that only chaos can cope with chaos.  But, more than
that, only chaotic dynamics can give us those great leaps, jumps,
twists and turns in human history that mark invention, evolution,
mystery, miracle and discovery.  One is not to fear disorder in the
heart of human institutions but rather to understand it and,
perchance to embrace it as the source of whatever progress in human
affairs awaits the knowledge process in the 21st Century.  The
first step in this journey to new understanding is an understanding
of the new geometry of process and structure revealed by this
perspective: the attractor.

Attractor:  A region in an outcome basin to which the dynamics of
     a system tends to take it.  

     In charting the dynamics of any system in time-space, a
special technique called a cartesian graph is used.  It is called
cartesian after René‚ Descartes, a french philosopher, who developed
it as part of what is called analytic geometry in 1637.  In the
literature it is simply called phase-space since such a graph shows
the changing phases of some event in which one is interested in
time and space. One sets one's measures on one, two, three or more
axes of a chart.  You and I are familiar with 3-dimensional space. 
In practice, cartesian time-space could have any number of
dimensions...in fact, there are some theories in physics which use
up to 25 dimensions...a very complex structure indeed.

     There are three or five generic kinds of attractors depending
on how one counts; I use the four below since, for human purpose, the
differences between the torus and the butterfly attractor are of
great interest; more so than to mathematicians who count three. 

Figure 1: Two Views of Four Attractors

The first four I use begin with the point and the limit attractor which
I promptly set aside as irrelevant to actually existing dynamics; the
which are of most use to a postmodern criminology are the Torus, in Box C,
above and the Butterfly Attractor in box D, above.

There is a fifth set of Attractors, multiples of the Butterfly Attractor
which spin off into Deep Chaos.  I will use Butterfly Attractors with 2,
4, 8, and 16 'wings' for many purposes but it is well to remember that
all complex attractors are varieties of the Butterfly.

     The most interesting thing about a fractal is that its
geometry is fuzzy.  It does not have neat edges, smooth surfaces,
compact content or clear boundaries between it and the next
attractor.  Indeed, some attractors are so open that one or more
other attractors, entirely different can occupy the same time-space
dimensions.  

     You can 'see' two views of four of the five attractors in
Figure 1.  The fifth dynamical state can't very well be called
an attractor since it is so complex that a system could end
up an almost an infinity of places in time space.  The
uncertainty of fate which awaits people, companies, churches, or
whole populations is so great that the everyday concept of Chaos
seems apt: utter confusion, uncertainty and hopeless chance.

  The first three time series in Figure 1; Boxes A, B, and C, are
neat and orderly enough for us to see the patterns there without
difficulty.

Indeed, these attractors are the stuff upon which the neat and orderly
theories in modern sciences are predicated...certainty, prediction,
control and planning are possible in these attractors.  Not so , in those
which follow.

Looking at the fourth attractor, Box D, we see that
the time series is so complex, it is hard to imagine any kind of
order in it...it is when we turn to Descartes and his analytic
geometry when the picture resolves itself into a very definite
shape...one we can 'see' in ways just looking at the system itself
as it changes in time and space would not emerge.  Even looking at
a chart of its ups and downs, ins and outs, turns and twists, does
not help much for us to see its overall behavior.  The Attractor in
Box D is called a butterfly attractor since there are two distinct
'wings' or patterns it could make; two very differing fates await
the child, the family, the business, the church, the school even
when they are begin in the same initial circumstance!  These
dynamics become even more jumbled up as we approach deep chaos.

     The remarkable finding in Chaos theory is that these dynamical
state exhibit an elegant and well ordered progression from
certainty to uncertainty.  Figure 4, below offers a connected view
of how these attractors fit together in phase space.  That
progression is seen in what are called bifurcation maps since, at
each bifurcation, new outcome states emerge and uncertainty
increases by orders of magnitude.  Yet even in deep Chaos there is
order and thus a chance for social policy as we shall see in the Chipset
to follow.

Strange Attractors  The first two attractors, in Figure 1 are
called point and limit attractors, as mentioned, these are familiar; 
they fit the assumptions of modern science nicely.  It is the next three
which are strange to the assumptions of modern science from Newton to
Hawking...they comprise regions of increasing uncertainty.

Simple systems are predictable; complex systems are not.  This is very
strange to the research of a modern scientist; they simply don't
behave in the neat and tidy fashion that Bacon, Newton, Descartes,
and Laplace assumed held true for all systems in nature and
society.  The second two dynamical patterns in Figure 1, Boxes C
and D are strange in this sense.  They are less than predictable.

The torus in Box C exhibits fairly regular dynamics; we can always
find it someplace within the closed cylinder of the sort in Fig. 1, 
Box C. It embodies First Order Change.  We can know roughly where, in
phase space a system can be found but not with the precision required
by 'grand' theory.

The Butterfly attractor is even stranger since it embodies Second Order
Change...it can be thought of as two linked tori...and greater uncertainty
develops when a torus bifurcates and becomes a Butterfly Attractor. The
dynamics of a system or set of systems alternate between states 
very different to each other.  This sort of change in dynamics was 
altogether unexpected when Ed Lorenz found it in his research on weather 
systems in 1962.

It is at this fourth bifurcation that we find Third Order Change; change
so rapid and confusing that science as we know it becomes impossible. But
not all is hopeless...there is much order in Region D and, with new
techniques now developing, it is possible to find attractors hidden deep
in Region D.  Patti Hamilton at Texas Woman's University has found such
attractors in complex data sets from teen-age birthing.  Figure 2 shows
the hidden Attractor Hamilton found.

Fig.2. A Hidden Attractor. Hamilton, 1994

Human Agency.It is very, very important to note that Region C, in Figure 2, 
is the preferred region for all social systems...Regions A and B contain so
much order that flexibility, adaptation, and creative change are not
possible...Region D contains so much uncertainty that human agency is
impossible...[double click on the Heading for more about Chaos and 
Human Agency].

Deep Chaos. Region 5 is the region in phase space in which dynamics are very chaotic.
When a fourth bifurcation occurs in any key variable, the number of end
states that a system might take becomes great indeed.  Third Order Change
develops after a Butterfly Attractor transforms from a dynamical state
in which there are 16 linked tori to one in which there are 32 linked tori.

Ever Stranger and more complex attractors follow one another rapidly.
Prediction, Certainty and Control are now longer possible.

 
As one moves from Region 1 to Region 5, Order decreases and disorder 
increases.  Later on, we will find that there are very specific points 
at which these bifurcations occur to alter the ratio of order and 
disorder in any given dynamical regime.  

These bifurcation points are very important to the possibility of social 
policy in keeping or changing a given dynamical state. They are called
Feigenbaum points.

CASCADING ATTRACTORS.  It is one of the most interesting features
of this science of complexity that it is possible to have more than
one pattern which describes the behavior of any system or set of
systems.  This does not seem to be paradigm shaking until one
realizes that, in normal science and given the same factors, there
is one and only one 'natural' or 'normal' pattern to which a system
is 'attracted.'  What makes this new science complex is that there
is a sequence (cascade) of the dynamical patterns a system [with
exactly the same set of parameters] could take.  Think of
it...instead of being able to predict the outcome or fate of a
person, firm, group or society by knowing all the parameters in
precise detail which shape its behavior, there are three kinds of
dynamics in which this is not possible.  Prediction fades and fails
at the edge of Chaos.

Fig.3 A Feigenbaum Map Showing Bifurcation Points

There are several important aspects of the Feigenbaum Map in Figure 3
to note and keep in mind as we create a postmodern criminology.
First, one can make out four bifurcation points before bifurcation itself
is so complex that it is hard to track such points.

Second, the first two bifurcation points define a region of certainty;
it is this region upon which modern science so depends for its triumphs
and upon which modern science is so focussed for its quest for truth.

Yet, Third, one can see that most of the dynamics of any given system lays 
beyond those first two bifurcations...indeed, dis-order is the common lot of
complex systems; those having three or more key parameters.  The quest for
certainty become, in postmodern science, a quest for bifurcation points
and for the new attractors which emerge from new bifurcations.

Fourth, there is a small region between the third and the fifth bifurcation
which is the preferred region for human beings; it provides enough certainty
for purpose and planning which it provides enough change and flexibility
for adaptation and creativity.  Those who prefer order do so at the loss of
the most valuable assets any human, any group, any firm or any society might
have.

Orders of Change    The torus represents first order change since
the system always takes a similar but not precisely the same
pathway in the same unit time.  Point and limit attractors always
that the same pathway hence exhibit movement do not exhibit change
in the technical meaning of the term.  As we shall see, some
attractors are far more open...they don't return to the same end-
state each cycle.  They may fluctuate between one end state and
another; this variation in outcomes is found in the butterfly
attractor, below and constitutes second order change.  Third order
change is found in the creative ferment of deep chaos...all
differing orders of change can be seen in Figure 3, below.

THE SECOND CHALLENGE    The first challenge to postmodern criminologists
is to learn the basics of Chaos theory...above.  The second challenge
is to begin to apply the new body of knowledge to criminal behavior.

The first task in that enterprize is to think about why people 
move from one way of getting resources to another way; that is to say, 
when and why do new attractors arise in a causal field with the same 
variables.  Or why people move from pro-social behavior in dealing with 
other people to another, more violent way.  

It turns out that, given a slight increase in a common 
ordinary parameter/variable, entirely new ways of behavior
arise; some of which we call crime. Crime emerges from the
ordinary, everyday workings of societies.

     If we can find these hidden patterns, then we can experiment to see
how and when small changes produce such bifurcations in human
behavior since the size and shape of any strange attractor depends,
sensitively, upon key parameters and the dynamics to which it is
driven by such parameters.  And, although there are limits to human
agency in such situations, still there are moments when very small
adjustments might prevent or stimulate the kind of attractors which
benefit individuals and societies alike.  In this limited intrusion
into the dynamics of strange attractors is the possibility of
change and renewal not possible in the fixed and certain worlds of
modern science.

     In the case of fish, bird, and insect populations, weather and
competition for food are key parameters which shape and preshape
the pattern/attractor.  In the case of the burglar, a wide variety
of parameters merge to give rough and uncertain similarity to the
behavior of our man.  The range of needs and desires for resources,
the number of potential victims, the kind of goods found inside a
house as well as policing patterns, themselves taking the shape of
a strange attractor.

     So, instead of one kind of behavior produced by a set of
variables, there may be up to five generic attractor states with
which to describe the pattern of behavior of any system [pendulum,
bird, star, person, society].  The first two attractors are not ordinarily
observed in nature or society; they can be found if one controls
all but one or two variables but in the doing, much of interest
escape human understanding.  

It is the last three attractors which are of great interest to 
postmodern criminology; indeed to every social scientist.  
We will cover these in far more depth later but,
right now, it is important to get a basic idea of each kind of
attractor and tie it more closely with human behavior generally and
crime in particular.

BASIC CONCEPTS IN POSTMODERN CRIMINOLOGY.  In later lectures, I will
try to be a good deal more analytic in the effort to build a post-
modern criminology based on the new sciences of chaos and complexity.

Just now, I want to hint at some of the applications to which basic
concepts might be put. We will start with the most basic concept of 
them all:

     Attractor, Point:  The pattern of behavior of a system whose
     dynamics tend to converge to one point in phase-space.  A
     favorite example is that of a pendulum which tends to revisit
     a given point at precise intervals or to come to rest at a
     specific point each time it is perturbed.

     If a professor were to say exactly the same thing at exactly
the same time in the first class meeting in a criminology class in
every semester, his/her behavior would be described by a point
attractor.  If a shoplifter were to visit the same store at the
same time each day [or week or month], his/her behavior would
describe a point attractor.  That doesn't happen unless one is
using video tapes; so point attractors are important to scientists
other than behavioral scientists.

     Attractor, Limit:  A very stable pattern of behavior in which
     a system moves between two points of which an automobile
     cruise control or home thermostat are examples.

     If a parent were to require children to go to bed after 8 p.m.
but before 9 p.m., their bedtime behavior could be described by a
limit attractor.  If a bad check artist were to write her/his
checks for no less than $450 and no more than $485 [the amount many
stores set for acceptance of checks with purchase since weekly
payroll checks for most workers are seldom more than that], then
that behavior would take the form of a limit attractor.  That
happens but, more often human behavior is a bit more irregular so
the torus is much more likely to be seen in the search of any given
data set.  While the point and limit attractor are well within the
logics of modern newtonian science, the torus gives us our first
view of a stranger attractor.

     Attractor, Strange:  An attractor is simply the pattern, in
     visual form, produced by graphing the behavior of any system. 
     A strange attractor is simply the pattern, in visual form,
     produced by graphing the behavior of a nonlinear system.  If
     the dynamics are likely to be confined to identifiable regions
     somewhere in a region of phase-space, it is said to be
     attracted to that shape.  Since that behavior of a nonlinear
     tends to be both patterned and unpredictable, it is called
     strange.  

     The point attractor and the limit attractor behave pretty much
as newtonian physics, Aristotelian logic and euclidean geometry
would have them behave...very orderly, very predictable, very
controllable.  The torus displays a little uncertainty; the
butterfly more and as more complex attractors develop, uncertainty
continues to increase and at the same time, change the shape and
orderliness of these attractors.  Beyond the edge of chaos, defined
by Region 4 in Figure 1 above, uncertainty evermore displaces
predictability; disorder evermore wins over harmony.  

The butterfly attractor is of great interest to the postmodern
criminologist in that it embodies the idea that a given set of human
beings suddenly behave differently.  The idea of the born criminal,
the idea of physiological sources of violence, the idea of bad homes
and failed mothers are set aside in this criminology in favor of
a small change in some key variable which triggers a large change
in murder, theft, arson, war, pollution, racism or some other form
of anti-social behavior.

DEEP CHAOS. The fifth regime of great concern to the postmodern
criminologist is that of Deep Chaos.  It begins at about the fourth
bifurcation. After that point, the cascade to great uncertainty begins
in earnest.  Modern scientists would give up in despair.  Not so
the postmodern scientist since the postmodern scientist has far different
epistemological aims from his simpler cousin.  There is order even in
deep chaos. The task is to find it.  There is the possibility of social
policy, the task is to specify it.

THE THIRD CHALLENGE: Research Objectives We will come back to these 
concepts in later essays in order to more fully explore more fully their 
use in criminology but for now, for the criminologist, the most interesting 
research questions lurking in such structures is how to find those hidden
attractors in raw data; how to locate the key parameters which produce 
those attractors; and how to find the bifurcations points which alter the
ratio between certainty and uncertainty.  Given those parameters
and an fore-knowledge of when new regions open up in an outcome
basin, we have the beginnings of social policy informed by
postmodern science.

     A subsidiary interest involves the ways in which contrary
behaviors occupy the same regions of phase space.  In complex
systems, as we shall see, prosocial behavior may be found in every
region in which antisocial behavior is observed; antisocial
behavior in every region in which prosocial behavior is observed. 
It is the changing mix of such behaviors toward which this new
science points us.  It does not serve to 'throw people in prison
and throw away the key.'  It better serves to alter the mix of
uncertainty to forestall behaviors we care not to experience in our
daily lives.

ORDERS OF CHANGE    When one leaves the well ordered realm of
point and limit attractors, one enters into the surprising world of
nonlinear transformations; leaps, twists, reverses, turns and knots
which cannot be followed by rational numbering systems or by formal
logic.  We approach the edge of this strangely ordered world of
nonlinearity when we follow the dynamics of simple systems.  The
first such attractor is a torus.  The torus becomes of more
interest to the criminologist since, on the one hand, causality
becomes fuzzy and uncertainty sets in as partner to predictability. 
Measures of correlation which require and look for a tight
connection between cause and effect lose epistemological utility in
sorting out the dynamics of crime, reward and punishment.

     Attractor, Torus:  An attractor created by the dynamics of a
     simple system driven by two variables and exhibiting one and
     only one loose cycle of behavior.

     Figure 4 gives a better view of the torus itself.  It is shaped a bit
like a doughnut.  A nonlinear system such as the torus is driven by two
interacting key parameters, above, feedback between which tends 
to drive the system in a loose and limited pattern.  One cannot say just
where the system will be found at any given time; all one can say is that it
will be somewhere on a pathway produced by two variable and interacting
parameters.  Such systems thus can be found anywhere on or in the cylinder of
the torus but cannot be found far outside the region occupied by the
torus.

Figure 3 shows the dynamics of a special view of the torus (from the 
Latin; a swelling).  I use the torus later on to explicate several kinds
of crime and deviancy, but for now, you can see that
it is mapped out on three axis which define three variables.

The key point to take in thinking about the behavior of any given individual,
any firm, any family or any group is that, while one may predict that the 
path of the system will be within the torus, it is impossible to say just 
where it would be.

This fact gives rise to the first significant feature of postmodern philosophy
of science.  Uncertainty begins to displace certainty.  I call this first
move to uncertainty, First Order change.  The Butterfly Attractor gives us
Second Order Change and we find Third Order Change in Deep Chaos.

 Figure 4, The Torus

First Order Change   A Torus is driven by two interacting variables.
There well may be hundreds of other variables in included systems
but they don't drive the variations of the system. In Chaos theory,
small changes in ordinary,everyday parameters produce First Order Change.

In the case of, say, an embezzler, the two key parameters
which might drive the patterns of his/her crime might be first, a
small change in demand for additional funds (say an unexpected medical
expense) and second, a grievance or slight at work (say s/he has been
passed over for a promotion in favor of a younger, less experienced employee in
a bank, county office or some other position of trust).  Given both the
perceived need for additional funds and some act which disenchants one in a
position of trust, one might well consider theft as a solution to this 
'problem.'

     In the case of medical malpractice, the two interacting
variables might be again, some uncertainty in expenses and again,
a position of trust in which opportunity for generating additional
funds might obtain. If, for example, an physician were to do
hysterectomies in a pattern which increased and decreased over the
year as seasonal expenses varied, such behavior would take the form
of a torus.  

For example, in Winter, surgical operations might be few
since weather and Holidays converge to divert doctors, staff and
patients; in early Spring, when doctors buy new cars, when income
taxes for self employed professionals have to be paid, when one
must buy an IRA and other self managed tax exempt annuities before
15 April, such operations might increase.  In summer time, real
estate purchases, stock investments and social rounds require extra
funds.  In the Fall, when children are off to expensive colleges
and universities, demands on income increase; revenue has to be
generated from some source.

     The same shift in ratio between income and expenses can arise
from fall in income as well as increase in expenditures.  Divorce,
stock market crises and accidents occur to alter the ratio between
income and outgo.  These attractors/patterns/rhythms might well be
hidden in the more complex data of medical practice and, in the
searching of such data with statistical tools, might escape
attention since the connections between income and expenditure is
so loose that correlation tables push them aside as 'non-
significant.'   Using new techniques for finding such attractors in
data sets, we might find a torus which tells us that, far from
being insignificant, there is a nonlinear but structured pattern
which could be modulated.  

     We be able to institute some oversight of the hysterectomies
a physician performs by comparing that pattern to a torus defined
by cycles of such operations by a reference set of physicians. 
Such unit acts, falling outside the boundaries of a larger torus
which described the operations of similarly situated physicians
well might be cause for some discussion among peers and concerned
parties.  The social policy implications are obvious.  A Medical
Board in a State or Federal Agency might thus exercise oversight
which serve the interests of patients generally and third party
insurers rather than dismiss such behaviors for 'lack of proof.'

In such simple dynamics, a doctor might well stray into illegal
and/or harmful behavior once in a while but when a torus obtains,
it is so seldom and involves so few doctors that a new and permanent
wing does not emerge. Figure 4 shows a tongue extending at the outer
edges of a torus. We can image a few doctors pushing the outer
limits of 'ethical' behavior then return to a form of practice in
which patient care, social honor and public trust drive the system.

Fig. 4. Pushing the Envelope

     The torus thus can be used as a research tool for any number
of simple behaviors the patterns of which might be hostile or
beneficial to the health or financial interests of patients and
third party carriers of insurance.  There is much that the torus
cannot do of greater importance to the knowledge process in crime
and social activity; for those purposes, we need to move closer to
the edge of chaos and make use of the Butterfly Attractor.

Second Order Change in Crime Rates.

     In the case of the point, limit or torus attractor, there is
one and only one general area in which in given system might be
found.  In the butterfly attractor, not only do we find nonlinear
dynamics but we find two outcome basins resulting from the same
initial settings, Figure 6.  This is strange indeed from a classical causal
model in which all similarly situated cases end up in the same outcome basin. 
Causality becomes complex indeed.  The implications of this for the knowledge
are fundamental; the same variables can produce two very dissimilar patterns of
behavior.  The implications for social policy are profound; for example,
rather than reducing criminal behavior, Chaos theory suggests there may be a
point at which punishment increases crime rates.  More about this in the
last Chapter; right now, let us look at the structure of this attractor.

     Attractor, Butterfly:  
	  A butterfly attractor has two outcome basins
     around which a system rotates.  The butterfly attractor can be
     thought of as two connected tori; at a given setting of a key
     parameter, an individual system could wind up in either wing
     (torus) of the butterfly.

     In point, limit and torus attractors, the existence of one and
only one outcome basin conforms to our expectation that all normal
systems behave 'normally' by doing about the same thing every time.  
In nonlinear dynamics defined by a torus, they seldom if ever do exactly
the same thing twice; self similarity displaces sameness as the nature
of natural and human behavior.  However, we could explain away the
variations in research findings by appealing to some putatively
missing intervening variable; by scoffing at the poor measurements
a research scientist made; by claiming that with more precise tools
we could get better data or by pushing aside contradictory findings
as evidence of bad theory.

Not so in the Butterfly Attractor and its more complex cousins. There
are two [or more] distinctly different patterns which all ordinary
systems may embody. The same variable produce both crime and normative
behavior.  The same variables are predictable and manageable at one
setting of a key variable and become less predictable and less manageable
at a second setting.

In this sense, all science is reunited since in nature as in
society, strange attractors are also common.  Grounds for the
separation between natural 'wissenschaften' and 'sozial
wissenschaften' inserted by 19th century philosophy of science
disappears at the edge of chaos.  The first inklings of this
commonality (there remain important differences), is found in the
torus but even more visible in the butterfly attractor since the
butterfly attractor has two outcome basins.

It is in the Butterfly Attractor that we first meet the great change
in human behavior that displays semi-permanent, semi-stable patterns
of crime. In the case of the torus above, most behavior is predictable:
i.e., most behavior fits within normative rules for, say, medical practice.

2n Basins: The butterfly attractor is most interesting to
the criminologist; indeed to every behavioral scientist for a more
practical reason.  Here, for the first time in the philosophy of
science, we see empirical grounds for a theory of normal/deviant
behavior which accepts that there are two outcome basins, Fig. 5,
in the larger outcome field which are equally natural to the behavior 
of a system.  I will pick up on the plurality of quite ordinary
variations within a given causal complex in a postmodern theory of
deviancy later, but for now, let us focus on the meaning of
the bifurcation (forking) of an attractor for the genesis of
criminal behavior itself.

Fig.5 The Butterfly Attractor

     If we draw a cartesian map in which prosocial behavior is
charted on the left wing of the butterfly while behavior on the
right wing is anti-social--then, in logging of the unit acts of a
given individual over time we can see that the same individual,
with the same socialization, and with the same set of circumstances
may fluctuate between very helpful behavior and very harmful
behavior.  

     The geometric ratio between a wing depicting anti-social
behavior on the one side and prosocial behavior on the other can
vary.  It varies by virtue of the objective circumstances of the
person, the firm, the group or the nation in the larger social
order.  Thus, the science of complexity is a science of the whole
rather than of the parts.  One cannot sort out the origins of most
criminal behavior by focussing upon genetics, psychology, or body
chemistry.  And if people are mad when they kill wives, presidents
or Popes, the prior question is why this madness takes this pathway
rather than visit more harmless basins.

     Robin Hood banditry offers this kind of bifurcated dynamics. 
The same individual(s) acts kindly and offers support for some
persons while cheating, stealing, robbing others.  Thus a
kidnapping in Italy may redistribute wealth from wealthy landlords
in Rome to a poor village in Sicily.  A prostitute may be cheating
clients, dealing in drugs or shoplifting during the day while
supporting his/her children and other kin at other times.  A
pharmaceutical firm may add to the infant mortality rate in Puerto
Rico by its polluting practices yet may dramatically lower the
death rate for cancer patients in Oklahoma by importing drugs at a
cost low enough to satisfy stock investors.  The geometry of good
and evil is not simple in the postmodern paradigm informed by Chaos
theory.

     In any criminology oriented to the logics of Chaos theory, one
would look for key parameters which promote or discourage such
bifurcated behavior. In his excellent article on banditry, Pat
O'Malley of Monash University in Australia found that:

     *bandits were supported by their rural community
     *bandits tended to rob class enemies: merchants and squatters
     *bandits tended to redistribute wealth downward
     *bandits are symbols of resistance against the ruling class

     O'Malley confirmed Hobsbawm's theses that banditry disappears
when the state ceases to act on behalf of class elites.  Hobsbawm
showed that unorganized class conflict is the key parameter for
banditry.  When the urban and rural poor are able to organize for
social justice in institutional politics, such crime declines.  One
would expect that most such rural people would be prosocial most of
the time and even in an exploitative situation, not engage in
banditry.

     In explaining the origins of banditry, Chaos theory would lead
us to look at, say, the percentage of wealth appropriated by feudal
elites, by class elites or by occupying armies; the percentage of
persons with political franchise; the ratio of ethnic or religious
minority to majority in 'racist' societies or, perchance the ratio
of such groups in low and high status occupations.

  Basin:  The region(s) in an larger field of outcomes toward which
     a set of initial conditions patterns, i.e. 'causes,' the a
     system or set of similar systems to move.

     Image a saucer inside which spins a marble.  The path of the
marble is the attractor; the whole region is the basin which
confines the attractor and the marble could go any where inside the
saucer as long as it is moving.  If the path is a nonlinear
function of two key parameters, the marble never takes quite the
same path twice.  Depending upon scale of observation, much or
little space is visited by the marble but never does the marble
visit each and every point on the surface of the saucer, therefore
that area can be considered a fractal causal basin; the marble
visits only a fraction of the space in the saucer available to it. 
Even if we could keep the marble spinning for ever, still there
would be points on the surface of the saucer not visited...the
unvisited regions would get smaller and smaller as a portion of the
total space inside the saucer but never completely disappear.

     This fractal character of an outcome basin is most trenchant
to a theory of truth.  In modern science, truth statements are
valid if and only if they are absolutely true.  In this new science
of Complexity, truth statements themselves are fractal since the
field they depend upon is fractal.  In modern science, we have to
wait for a 'strong' correlation before we can ground social policy
on, say, low level radiation, on cigarette smoking, on toxins in
food and water supply, on television and pre-teen violence or any
number of factors which are said to do harm to body and spirit.  In
such a science, the very notion of crime is changed from an
absolutic concept to a more nuanced one with infinite shades of
grey.  Verdicts of guilty and not guilty are far too binary and
mutually exclusive to capture the complexity of crime in postmodern
criminology.

     The butterfly attractor brings with it two outcome basins in
an outcome field.  Given a small increase in a critical variable,
a person, business, church, or society might take a qualitatively
different pathway in response.  I have mentioned some of the more
arresting implications of this twinned but different outcome field
for philosophy of science.  For criminology it means that more of
the same can force/push/make desirable completely different ways
of behaving.  A slight increase in taxation can make corporate
officers reconsider criminal actions unthinkable in a 'more
favorable' investment climate.  A slight increase in frequency of
battering by a spouse and produce a qualitatively different
response on the part of the abused person.  A slight change, in
short, increases the ratio between order and disorder in the whole
outcome field.   A person, corporation or social group can no
longer be depended upon to act in habitual ways.

     The appearance of new outcome basins in an outcome field is
explored, in the science of complexity, by bifurcation theory.  Why
and when do these slight changes trigger large transformations. 
The when is well known; the Feigenbaum points discussed below
identify the onset of bifurcation(s).  The why is known in
mathematics and in thermodynamics...it is not known in social
phenomena.  In that profound ignorance is the challenge of the next
generation of criminologists in particular and all behavioral
scientist in general.  Let us consider the production of new and
different patterns of behavior through bifurcations.   

Bifurcation Theory.  You have seen the bifurcation map above in
Figure 3 and can see that there can be any number of such
basins/attractors in a larger outcome field depending on how many
bifurcations have occurred.  This is where the criminology informed
by Chaos theory departs most dramatically from that informed by the
newtonian paradigm.  In nonlinear dynamics, it is possible that the
same variables involving the same systems produce two or more very
different outcome basins.  Rather that a set of parameters
predicting on one and only one outcome for a given form of
behavior, a slight change in a key parameter can produce a large
change in behavior some of which is criminal while large changes
might not produce any significant change (this is the nonlinear
feature of such dynamics).  More interestingly, another still small
change can double again number of outcome basins to which that same
set of systems (no change in them) would go.  This is the single
most remarkable finding in Chaos theory.

Bifurcation  A doubling in the pattern of behavior of a system. 
     With each doubling, there is distinct change from one
     behavioral regime to new pattern(s) for all systems.  Small
     changes in one or more key parameters produce that doubling. 
     After the third bifurcation in key parameter(s), the system
     tends to move in ways which fill the space available to it in
     an outcome basin.  This latter state is a far from stable
     chaotic state.

     In order to see the beginning of a butterfly attractor just
before a bifurcation of a torus, we will take a cross section of
     Figure 4, above, shows a cross-section of a torus (called a Poincare'
section) in which one can see more closely that tongue which marks
the advent of a new outcome basin.  We will learn later that such
tongues expand into their own outcome basin at specific values of
key parameters.  Bifurcations occur with singular, regal
regularity.  The procession of bifurcation points (called
Feigenbaum points after the physicist who discovered them; see
Gleick, 1988:157).  Identification of the Feigenbaum points offer
the greatest challenges to postmodern criminologists.

     There remains much pattern and predictability after the first
three bifurcations; at the fourth bifurcation of a key parameter,
there is a cascade of bifurcations such that tiny increases in key
parameter values trigger very unstable behavior.  Identification of
key parameters and discovery of the bifurcation points is a major
challenge for postmodern criminology.  It is a curiosity of Chaos
theory that even with such increases in uncertainty, still there is
so much order and, with order, science is possible; prediction of
a sort is possible, truth statements are fractally possible, and
for human purpose, control and planning are still possible.  

     The concept of bifurcation is most important to the
criminologist in that it sensitizes one to expect the emergence of
an alternative way to do business, family, religion or perchance,
crime given a small increase in a key parameter.  In a concrete
instance, a firm which for the past 70 years has managed an
employee's pension fund with honest agency, may raid it as a result
of a small change in the interest rate, the tax rate or a law suit. 
At some point, empirically set, an small increase in taxes,
interest, prices or living arrangements will produce a qualitative
change in the number of people committing crime...conversely, at
some point, to be empirically discovered, a small decrease in
wages, salaries, pensions, profits or other income might do the
same; produce qualitative increase in the number of persons doing
criminal acts.

     The criminologist would do well to conceptualize theft,
prostitution, conversion of property, price-fixing, genocide or
fee-splitting as outcome basins to which people and firms move as
a consequence of small changes in external constraints,
opportunities or, perhaps, internal needs or desires.  Given such
a view, the multitude of parallel economic activities, some of
which are prosocial, some antisocial, can be seen to be adjustments
people, firms and societies make to small changes in the larger
whole in which they are found.  Given such a view, recourse to
violence, self destructive behavior or hate crimes can be seen to
be adjustments permitted/obstructed by the larger structures in
which a person or a group find themselves.  Thus one need not have
a theory of crime per se, in order to understand why new but anti-
human behavioral patterns arise; one needs a theory of the
interactions between needs, wants, goals and cultural imperatives
on the one side and constraints on the number and accessibility to
given basins on the other side.  A theory of crime is thus also a
theory of alternatives in a changing mix of order and disorder.  In
such a view, crime and its dynamics are relocated from the separate
person of the party involved to changes in constrains within the
larger environment.

     Genes, drives, physiology and psychology of the acting
individual are essential to all behavior.  Labeling and societal
reaction are essential to all human behavior.  Differential
association is essential to all behavior; both social and harmful. 
Controls are essential to all behavior from talking to warfare. 
Habit, addition, passion, and desire are common to all forms of
human behavior.  When we speak of crime, we speak of patterned
behavior.  When we speak of patterned behavior, we speak of the
parameters which limit and facilitate that behavior.  Those
parameters are the proper concern of postmodern criminology and,
indeed, all behavioral science.  The interesting question becomes,
what are the points on those key parameters, beyond which anger,
violence, oppression and exploitation are directed at innocent
others.  For social policy, the complementary question becomes,
what are the settings of key parameters which maximizes prosocial
behavior and minimizes behavior hurtful to others in a socio-
cultural complex.

     Whatever constancy and predictability which obtains within an
causal field is found in causal basins produced by those nonlinear
transformations; not in any unchanging connectedness between
dependent and independent variables as presumed in modern science. 
Thus poverty could produce sharing and helping at one setting of
another variable and, with a slightly differing setting degenerate
into theft and violence.  Causality opens, closes and transforms in
this paradigm.  Contradictory findings; inconsistent findings and
changing correlations are commonplace as bifurcations ensue.  The
relationship between variables changes as one samples different
regions in an outcome field.  In some basins, causality is fairly
tight and positive; in other basins, it is fairly tight and
negative; in the regions between basins of a butterfly attractor
certainty yields to surprize, creativity, and wonderment.

Dynamical Key:   Each attractor has its own characteristic set of
     complex cycles.  If one can identify that set and match it
     with countervailing input, then it is possible to affect the
     behavior of the system (Hbler, 1992:18).  In theory, chaos is
     manageable.

     Far from being unmanageable, Chaos is in fact manageable
(Hubler, 1992; Young and Kiel, 1993).  Hubler says that, in the
management of Chaos, the more unstable the causal field, the
gentler must be the touch in trying to control the dynamics of the
phenomena in question.  Heavy handed control tactics might work
with point, limit and torus attractors; such tactics might work in
regions of a butterfly basin but, for causal fields with 4, 8, or
16 attractors, control efforts lose efficacy.  More on which later
in the Chapter on Chaos and Social Control.

     In criminology, if we are able to identify the key parameters
which drive a system and match the complex cycles of those
parameters with unwanted nonlinear response, it is possible to
maintain an uncertain stability even in deep chaos.  More feasible
for postmodern criminology is the insight that perhaps social
justice is preferable to criminal justice.  It may well be the case
that small change in cultural and economic parameters can increase
or decrease theft or violence.

It might well be the case that small increases in unemployment may
drive large changes in burglary, car theft, robbery and domestic
abuse.  It well may be the case that small changes in tax laws may
produce great differences in wealth such that the political process
becomes a commodity bought and sold to the wealthy...while poor
people are reduced to futile anger...or underground rebellion and
resistance to the taxing authority.  

At the same time, it may be the case that building of more courts,
great prisons and introduction of evermore efficient policing
technology only adds to the population of those engaged in criminal
behavior.  It is far better to prevent rape, burglary, theft,
murder, pollution, embezzlement, or warfare than to watch every one
ever more closely or to put ever larger segments of the population
in prison.  The research capacity needed to provide information
about such dynamical keys is a matter of national and international
importance and should take priority over super colliders, nuclear
weaponry, arms production or military ventures in the 3rd world.

Chaos and Causality.  In non-linear dynamics, causality fades and fails.
indeed, it turns out that the concept of feedback is a better concept
to use than the concept of causality. 

Feedback:      When a system acts in such a way to affect other
     systems in a causal field, and when those systems, in turn,
     affect the behavior of the first system, a feedback process
     occurs.  There are three kinds of feedback of considerable
     interest to behavioral science and criminology; positive
     feedback, negative feedback and nonlinear feedback.

     In biology, the number of trout affect the number of pike
which, in turn, affect the number of trout since pike are predator
to trout.  The number of pike do not affect the stability of the
population of trout until that number exceeds the first Feigenbaum
point; with each successive bifurcation, causality increases until
the trout population is very sensitive to further changes in pike
population.  Causality is variable and a function of the number of
bifurcations of key parameters.  In human biology, a person can be
host to billions of pathogens and, with some small reduction in
resistance or with some small added increase in, say, exposure to
low level radiation, entirely new causal dynamics develop.

   In sociology, a symbol used by one person will often elicit a
similar response in another person whose response in turn with
elicit further response from the first person.  A greeting is a
case in point; if one says, Hello to another and if the greeting is
returned, both parties are open to further interaction.  If the
greeting is not returned, then feedback stops.

     There are three forms of feedback which are of interest to
Chaos theory: positive linear feedback which tends to push a system
into full chaos; negative linear feedback which tends to draw a
system down to a point attractor; and nonlinear feedback, which tends 
to maintain an unstable system in a given pattern.  

This last form of feedback has most interesting meaning for both the 
management of chaos and for social policy of any society which wants 
to maintain enough disorder to permit adjustment to a disorderly 
environment and enough order to permit prediction and control.

     For the criminologist as for the public, the most interesting
answer one can give in aid of domestic tranquility is what kind of
feedback is effective in creating a low crime society.  One answer
to this question is offered in the pages which follow; in brief,
social justice is preferable to criminal justice since social
justice is based upon mercy rather than rational application of
either market logic or penal logic.

It turns out that mercy, forgiveness, tolerance and aid are non-linear
responses to crime.  It well may be the case that crime is minimized
by pro-social response to the crime rather than anti-social infliction
of pain, degradation and discomfort.

Conservative will point out, correctly, that many criminals do not
respond to kindness, forgiveness, and constructive help.  That many
criminal feign repentance, rehabilitation and remorse...and laugh at
the people who believe that fraudulent dramaturgy.  But the larger
policy question is what form of feedback minimizes crime.  In fact,
hurting the criminal who hurts victims is positive feedback...it adds
to the overall burden of pain and suffering in a society.  If the
punishment fits the crime, it is negative linear feedback. 

The Fractal is another most challenging concept for criminology. It well
may be the case that every one engages in behavior defined as crime; that
when we look at all the billions and billions of unit acts in the life
of a person, we find scattered around in those acts, some acts which are
decidedly anti-social; racism, sexism, anti-Semitic behavior as well as
ordinary theft, fraud, embezzlement, conversion, arson and such.

Yet for every living human being, by far the most acts in that larger set
of human activity are pro-social...are normative.

We need quite a new concept to cover the changing mix of pro- and anti-
social behavior that is the life of everyone.  The Fractal serves well.

Fractal:  From the Latin, fractus, broken (frangere, to break). 
     A measurement of the degree to which a body takes up space
     available it; an estimate of its efficiency in using the space
     it occupies.  In more simple terms a fractal is a measure of
     the irregularity of an object. After Mandelbrot.

     Every nonlinear attractor has a fractal value.  Attractors
with low values occupy but a portion of the space available to
them; those with high values, occupy all the space available to it. 
A point attractor has a higher fractal value than a limit attractor
even though a limit attractor occupies a larger volume of space
since the limit attractor uses a smaller portion of what space it
does occupy.  The higher the fractal value of an attractor, the
less uncertainty, the more predictability and the greater the
possibility of control.  If one looks at the portion of space
available in an outcome field with, say, 4 attractors, there are
many places where one could find a given system at a given moment
and few places where one actually finds that system.  The problem
arise due to the fact that one can not be sure just where, within
that region bounded by the geometry of the fractal, that system
will be found.

     In terms of, say, domestic violence, one could be sure that
there would not be much domestic violence at given levels of
employment; and perhaps psychological variables would be most
helpful in sorting out as between those who do beat a spouse or a
child and those who do not.  At higher levels of unemployment,
domestic violence explodes while the possibility of predicting who
will be violent and who not, fades and fails.  In terms of theft,
if we consider all the millions of economic unit acts in which any
single person engages in the course of a month (from eating to
mending to selling refrigerators to buying a stamp), we might see
a stable pattern of theft with low levels of stress but as the
forms of stress increase, the number of acts of theft might take up
a much greater portion of the totality of economic acts observed. 
Theft has become a non-linear hence much more open fractal value of
all such economic acts.

     The sources of stress are many; all become stress in that
there is uncertainty in how to deal with budget problems, health
problems, marriage problems or problems at work.  It is not the
amount of problems which produce stress but the uncertainty in how
to deal with them.  A criminologist might do well to consider the
ways in which uncertainty contributes to crime.  In the life space
of a firm or a person, uncertainty in one key parameter might not
push one into criminal activity; a person might be able to manage
two interacting uncertainties but, if a third uncertainty occurs,
a person or a firm may opt to move to a more certain line of
activity; some of which may be criminal.  In general, if there are
solutions, then problems are manageable thus routine.  The
operative point here is that, if we want to keep crime stable and
thus controllable, we must consider policy for the kinds of
uncertainties that students, spouses, doctors, brokers, prostitutes
and priests must deal.

Postmodern Philosophy of Science and Knowledge.

Modern criminology assumes linear causality.  Chaos and complexity
theory posits a changing mix of order and disorder.  The first three
attractors in Figure 1, above, have enough linearity to satisfy most
modernist philosophers of science. Those attractors which follow do not.

Linearity become victim to bifurcations; causality becomes useless as
a concept.  Let us look at the larger task of the criminologist; let us
look at the reason why all criminologists in the 21st century will be
postmodern criminologists...why they use research techniques which presume
non-linearity and which are helpful in the hunt for hidden attractors.

Linearity    A linear system is one in which cause and effect are
     related to each other in a proportional way.  For instance, if
     one pound makes a rubber band stretch twice its length; a two
     pound weight will make it stretch four times its length.  For
     many systems, causality is curvilinear; a given cause will
     have an effect that is smaller with each additional doubling
     of it.  For many systems, the addition of the nth doubling
     will cause it to transform into non-linearity.  Nuclear
     fission is linear up to the point of a critical mass. 
     Nonlinearity marks the onset of chaos.

     Modern science presumes and privileges linearity; Chaos theory
presumes observes and reports the peculiarities of nonlinearity. 
In linear dynamics, the variables which produce crime are assumed
always to be present while it assumes the connection between
variables and outcomes always stable in good theory.  Not so in
Chaos theory.  The enduring presumption in Chaos theory is to
expect turns, twists, jumps, reverses, and wild swings which leave
much theoretical territory empty of theorems, propositions,
hypothesis, and other spendable scholarly supplies.

     If a criminologist were to use the Chaos paradigm for research
questions and causal inferences, one would expect to find changing
configurations in causal patterns depending upon both the stage of
bifurcation and the region of an outcome basin in which the data
were taken.  That means, for example, that poverty might be
associated with prosocial activity self-similar dynamics but be
associated with criminal behavior in a nonlinear fashion in more
complex outcome fields.  That means that disemployment may be
correlated with street crime in one historical epoch but not in
another.  That means that tough sentencing practices may reduce
corporate crime when tax rates are low but have no effect with even
a small increase in those rates.  Nonlinearity forces the scientist
to look for pockets of order in a larger sea of varying disorder
for those kind of precise statements so dear to the soul of the
counting, calculating, controlling metaphysician.

Nonlinearity:  A pattern of behavior in which a change is out of
proportion to the value of a variable in a causal matrix.

          There is a lot of pattern, predictability and thus,
     potential for control in self-similar dynamics.  In the class
     room, in the hospital, in the shop, factory and office, the
     routines of the day are similar from day to day, week to week
     and from year to year.  However small changes work to modify
     routines over the years so that self-similarity of this day
     may be qualitatively different from that of the next year. 
     With research designs which presume eternality in causal
     matrices, such qualitative change is lost to the presumption
     of faulty research design on the part of the prior generation
     of observers.  How to tell an erroneous false negative from a
     true false negative is added to the epistemological process in
     non-linear dynamics.

     In criminology, the supply of police officers might or
     might not affect the population of criminals.  The harshness
     of sentencing might or might not affect the length of the
     criminal career of the white collar thief.  Constitutional
     protections against state crime might work at one time to
     constrain officials who use their office on behalf of an elite
     inside or outside the nation but those same guarantees can be
     systematically set aside with great public acclaim at other
     times.  The fact that there are two contradictory outcomes,
     rather than one dependable outcome basin to which existing
     conditions can drive a firm or a person is daunting to those
     who prefer a more amenable data set.

Research Challenges for Postmodern Criminology.

In criminology as in all social research, the actual structure
of the underlying social and psychological reality is central to
the mission and the method of the knowledge process.  If one
researches, say the relationship between disemployment and crime,
one might find any number of relationships depending upon; a) how
one conceived variables; b) the region of an outcome field chosen
to study; c) the stage in a bifurcation map of disemployment
patterns as well as, d) the scale of observation.  All this makes
the research process much more complex on the one hand and much
more a human product on the other.

     The postmodern criminologist will want to build cartesian maps
of time series data in order to reveal the number and fractal value
of the hidden attractors in crime data for street crime, white
collar crime, organized crime, corporate crime and state crime. 
Once one has time series data in suitable form, there is a software
which one can buy to look for such hidden attractors; it is called
CDA, Chaos Data Analyzer and is published by the American Institute
of Physics as part of a continuing series of software from Physics
Academic Software.  The software package works for IBM PC, XT, AT,
and PS/2 computers.  It was developed by Julian Sprott at Wisconsin
and by George Rowlands at Warwick, England.

    A postmodern perspective fits excellently well into the ontological
paradigm defined by nonlinear dynamics since there are no centers,
absolutes or final states to which systems 'naturally' evolve.  For
those who need a theoretical world view with which to ground
plurality in cultural forms, variety in sexual experience,
diversity in religious sensibility, creativity and surprise in art,
drama, poetry and music or contrariety in economic systems, Chaos
theory provides that theoretic envelop.  For those who just want to 
be creative and unpredictable, there is little need for theory.

     A postmodern criminology must, first of all, review
definitions of normality and deviancy scrupulously in order to
avert the postmodern critique which finds so much such definitions
a politics in which one social life world is privileged over
another by a putatively neutral science.  It is not that
definitions of deviancy are not then possible; rather it is that
one must accept the political character of such definitions if one
is to be true to a postmodern sensibility.

     Rather than expansion of the criminal justice system,
postmodern criminology will find itself far more interested in
controlling the key parameters which enlarge the field of crime;
enlarge the population of those using this kind of activity or,
perchance in controlling the levels of desire which fuel criminal
behavior in doctors, accountants, merchants, bankers, and burglars
alike.

     Part of the postmodern knowledge process will depend upon a new
mathematics I have not yet mentioned.  It is very different from
the linear, rational mathematics of Newton and Einstein but yet
elegant enough in its own way.  John Briggs and F. David Peat have
a most engaging and accessible treatment of that math in their
Turbulent Mirror: An Illustrated Guide to Chaos Theory and the
Science of Wholeness.  (New York: Harper and Row, 1989).  Indeed
their treatment of many of the concepts presented here is a fine
place to begin to flesh out the content of Chaos theory.

     That math can give one a very close approximation of the
fractal geometry of the structures of crime in a society.  It can
predict with pin-point accuracy when one of the three great
transformations in system dynamics will happen.  It can find the
hidden attractors buried deep in a data set which looks random and
unpatterned.  It can separate noise from order and it can help us
gently and inexpensively design a bifurcation map that is more
amenable to human dignity and human agency than is now the case. 
And that is the best solution to crime; not a bigger and better
criminal justice system but rather a good and gentle society in
which the ratio of order to disorder serves the human need for
constancy on the one side and creative response to new conditions
on the other.

Hamilton, Patti, Bruce West, Mona Cherri, Jim Mackey, and Paul
Fisher.  1994. Preliminary Evidence of Nonlinear Dynamics in
Births to Adolescents in Texas, 1964 to 1990.  Theoretic and
Applied Chaos in Nursing.  Summer, 1994. 1:1