by Larry de Cléir
Mathematics is usually associated with the worlds of science, engineering, computers, etc., but it must not be forgotten that it also has applications in the natural world, i.e. the environment which surrounds human beings from conception to our final breath. The intention of this article is to provoke some thought amongst readers as to how knowledge of 20th century developments in mathematics, specifically Chaos Theory, has relevance in designing initiatives to protect children growing up in families who have been debilitated by many generations of involvement in addiction, criminal activity, and imprisonment.
Prior to beginning, there are two points worth noting: the daily routine of children in such families is frequently interspersed by crises, e.g. acute debt, frequent moves, fear of violence, chronic illness, sudden tragedies, unexpected losses, etc; the medication option is often the one offered by professionals to desperate parents wishing to ease their own and their children’s distress. It is my experience that for those children who grow up in families affected by imprisonment, the road to dependency; possibly depression and addiction, early school leaving; and ultimately crime and imprisonment, is often marked out, from a very young age, with shunting from expert to expert, from meeting to meeting, ingesting legally prescribed medication along the way, while watching the adults who have the most significance in their lives doing the same.
Chaos Theory and Advances in Mathematics in the 20th Century
Traditionally, mathematics ignored unpredictability, as its primary applications were the fields of science and engineering, where predictability and the ability to measure accurately were of vital importance. However, in the 20th century, mathematicians began looking at unpredictability and uncertainty in the natural world and over time developed theories which proposed that underneath the chaos, or unpredictability, that we frequently experience in our day to day lives, there is an underlying order, that is, a predictable pattern, that is not immediately obvious. The chaos that we are familiar with could be local, such as the day-to-day activities of our family, how a field is taken over by weeds, a series of minor accidents, or global, e.g. the spread of an epidemic or the incidence of earthquakes.
Readers may have heard of ‘the butterfly effect’: that is, ‘the flap of a butterfly’s wings in Brazil setting off a tornado in Texas’. A butterfly’s flapping wing represents a virtually imperceptible change in an initial condition of a climatic system, which causes a chain of events leading to a tornado. At some point along the course of events, the process reaches ‘critical mass’ that is, a ‘tipping point’ after which there can be no reverting to the original state. For further information I refer readers to a paper ‘The Butterfly Effect’ (Lorentz: 1963). Consider critical mass when applied to human behaviour. There is a saying amongst stand up comedians that if one-third of the audience laugh, all will laugh. We are all familiar with the phenomenon of the ‘buzz’, that is, we know something is ‘happening’ and we want to be associated with it. Why does a team usually play better at home?
All adult males will be familiar with, and may have been affected by, the ‘gang’ mentality in teenage years, that demands loyalty and can be very destructive. Sometimes a group of young men’s willingness to be violent reaches ‘critical mass’, seemingly due to the influence of each other as one or more, or all of them, do things that they might not have done had they been on their own. The ‘butterfly effect’ is observed here also, as the result of the incident, serious injury or possibly even death, is often highly disproportionate to the input, possibly an innocent remark or a mistimed joke. The event takes on a life of its own and proceeds with little relevance to the input. Somewhere along the timeline a ‘tipping point’ is reached where there is no going back to rational thought or reasonable behaviour.
I propose that on a journey towards a life of imprisonment, hopelessness reaches critical mass at some point in childhood where one forms the belief, at a deep unconscious level, that self-destructive behaviour has become self- sustaining, and that there is no ‘turning back’, no matter how one might coach oneself at a cognitive, conscious level to be better behaved to please others. On the other hand, I propose that responsibility can reach critical mass in a person who is, say, recovering from addiction. The person reaches the point when they are confident enough that they can live a drug-free, life- affirming existence, and that there is no going back. I believe that this can also happen in a family, or any group of people, but more particularly a small group, as responsibility, over time, becomes the norm where previously the norm had been irresponsibility. The important term to remember here is ‘self- sustaining’. A feedback loop of unconscious, or barely conscious, positive reinforcement of behaviour is enough to maintain the behaviour, as it, over time becomes the norm. This is further explored below when we discuss Chaos Theory and its applications.
Studies in early 1900’s physics, in particular, ‘Heisenberg’s Uncertainty Principle’ (on which much information is available from many sources), showed that the evolution of complex systems in nature cannot be accurately predicted. As part of his studies on weather patterns, Lorentz further developed theories on the incidence of randomness and chaos in nature, in particular the evolution of systems that can be deemed to be ‘complex’. I will define ‘complex’ here to mean a system that has many variables, that is; things that change over time. The family is an example of a ‘complex system’, as it contains many elements that vary over time. Two subjects are important in the ‘unpacking’ of the world of chaos. These are ‘fractal geometry’ and ‘self- similarity’. There are many examples of both in nature and often the growth of a tree is chosen as a typical example. We see a trunk, surrounded by sturdy branches, from which smaller branches emerge, with still smaller branches from those, and then twigs, smaller twigs, and then leaves, and within the leaves are capillaries from which smaller capillaries emerge and so on. The ‘fractal’ part is the breaking up of the main trunk into smaller branches and still smaller branches and then twigs and so on. The ‘self similarity’ is a symmetrical sequence where the overall pattern of the fractal is repeated at multiple scales. That is, a very small twig from which grows even smaller twigs will be similar to, but, and importantly, not exactly the same as, the trunk itself from which grow the branches. The ‘chaos’ part comes in when we consider that we cannot predict from looking at an acorn what exactly the oak tree will look like in a hundred years’ time. But we know that the oak tree will follow a particular pattern as it grows! Like snowflakes, all trees are alike but all are different also.
Relevance to Children’s Growth and Development
Applying fractal geometry to the influences on a child that is growing, we can analogise a ‘main trunk’ as being the child himself, i.e. his own sense of self; the sturdy branches as being influences of parents and/or main carers, then siblings; then the smaller branches as being, depending on their proximity and/or influence, grandparents, school friends, teachers, cousins; then with less influence, the twigs are the distant relatives, the community, town or city, the country, and so on. Each element has an influence on the child, but the more influential ones are the main trunk and thicker branches.
The self similarity occurs in the relationships. Similar dynamics will exist, and will probably have existed for a long time, within the systems that surround the child, though they will not be exactly the same. These provide a feedback loop that reinforces the behavioural patterns of the child over time. The child’s personality and/or identity, values etc. are all continually reinforced, and ultimately sedimented by virtually endless cycles of re-entry, feedback, provided by influential others. Like the tree, a keen and experienced observer might make a good guess at how a child will develop if they have knowledge of extended family dynamics, school ethos, community norms, etc., but the observer will not be able to predict exactly what will happen. Fractals, as they appear in nature, imply irregularity and change accompanying evolution and growth. An awareness of fractal geometry offers an understanding of how hidden connections and associations create predictable patterns beneath the unpredictability and uncertainty that we come up against in our daily lives.
When considering the family whose existence we might describe as ‘chaotic’, chaos theory would propose that the minute-by-minute encounters, events, incidents, crises etc. that the children in such a family experience can never be precisely anticipated. Yet, beneath the surface of the ongoing chaos will lurk that difficult to observe order that determines the general, long-term patterns of behaviour. And the existence of this often invisible order will also propose that it is not the one big traumatic event, e.g. experience of sexual abuse or death of a parent that makes the difference, important as such a traumatic event will be in a child’s life, it is the innumerable and consistent feedback that takes place over many years. Of course the corollary here is that the healing of trauma will not involve one big generous gesture, or a single brilliant intervention, but rather a consistent application of significant, warm and loving interventions over a long time – just like what would be happening in a healthy extended family!
From the 15th to the 19th centuries, science and mathematics challenged many traditional beliefs that had been held to be true for as long as anyone could remember. One example was the assertion that the Sun did not go around the Earth, rather, the Earth went around the Sun. It is difficult to appreciate now how unbelievable this was to people who lived in those times. The subjective experience of every human was, and is, to observe the Sun ‘rising’ in the morning, moving through the sky, and ‘setting’ in the evening over a stationary Earth. As the fact that mathematical equations could assert realities that contradicted subjective human experience filtered into the consciousness of ordinary people, it began to have an effect on our beliefs, norms and culture. The belief that science and technology could assist humans in exercising control over far more parts of our lives than in previous centuries became firmly established. Alchemy was replaced by chemistry. Astrology was replaced by astronomy. Superstition was largely debunked, and/or replaced by fact.
People began to believe that the chaos that is part of every human’s experience and caused so much trouble, e.g. plagues, famines, wars, could be eliminated from our lives if we so wished and replaced by certainty. This was very influential in many areas of life, e.g. engineering and farming, and we will focus on its influence on healing here. Advances in medicine based on mathematics and science had a significant role in eliminating chaos, or uncertainty, out of healing. This meant that rather than being a mixture of guesswork, heavenly invocation, and/or intuition, the exact reason why a person was unwell could be determined and the exact treatment that would make that person better could be scientifically delivered. Precise doses of chemicals could be given so that healing could be virtually guaranteed. Anaesthetics could be given so that a patient would feel no pain as a surgeon removed damaged parts of the body to provide relief, like a mechanic would remove a damaged part of an engine. It is no surprise that over the centuries the body came to be viewed as a machine of sorts, which could be ‘fixed’ just like an engine. This also, of course, involved handing over most of the responsibility for the healing process to an expert person who had studied various diseases and methods of healing in depth and knew exactly what to do.
Cures for the condition that was known as ‘mental illness’, however, continued to elude physicians. The results of dispensing chemical medication were patchy and brought little long-term relief. However, the ‘mechanistic’ model of healing continued to be very influential in developments to alleviate mental illness. The elimination of uncertainty and chaos became just as important in psychiatry and psychology as it was in physical healing. This is not surprising as order and structure are important in addressing emotional distress/mental illness anyway.
Chaos and Alleviation of Distress
Almost concurrent with the mathematical developments mentioned above, whether by accident or design, some practitioners in the mid to late 20th Century challenged the emphasis on logic and analysis in alleviating distress in persons suffering from emotional distress/mental illness. Carl Rogers, who developed Person Centred Therapy, is well-known as a practitioner who challenged the ‘order and structure’ healing paradigms in the mid 20th Century. Person Centred Therapy invites the client to bring his/her world into the therapy room and risks accommodating a considerable amount of discomfort, surprise, uncertainty, and even chaos. The unorthodoxy of Rogers’ work was much criticised in many circles, as it did not seem to be ordered into a beginning, middle and successful conclusion. The acceptance by the practitioner that a journey to wellness would involve the ‘patient’ presenting as ‘chaotic’ and thereafter making sense of his/her own erratic behaviour, and/or finding meaning in irrational reactions to events, with the practitioner as a ‘facilitator’, and not an ‘expert analyst’ who handed out a solution, was a departure from contemporaneous norms of helping and/or healing.
There is an element of paradox in all this, as it involves the introduction and inclusion of that which we are trying to reduce, i.e. chaos, but has a parallel in the world of physical healing, that is, inoculation. While Person Centred Therapy is now a well-established, respected and highly valued method of healing in the psychotherapy profession, I have not observed its considerable benefits to have reached, or be influential in, interventions, programmes, etc. that are aimed at the families described in the introduction. Indeed, I note that the vast majority of interventions, statutory and voluntary, that are set up to assist families whose children are at high risk of being involved in anti-social behaviour and serious crime, are largely influenced by the science of psychology, which is in turn strongly influenced by formal educational or medical approaches that, like engineering, are largely based on linear models, and see chaos as something to be abhorred! These may and probably do work very well for ‘mainstream’ families but, in my experience, they do not translate well to the children of families at high risk.
I observe the trend towards elimination of chaos more and more in recent years. A recent example that I can give is the changing of the name of ‘The Probation and Welfare Service’ to ‘The Probation Service’, which was accompanied by a considerable tightening up of the remit of Probation Officers in general. The Welfare was, after all, the bit that invited in chaos or uncertainty! Furthermore, I find that voluntary agencies that might have been traditionally oriented towards person-centred methods are under pressure to change also. I would say that it is now more difficult for a practitioner working in the area of child protection or criminal justice, whether statutory or voluntary, to do something spontaneous, innovative or creative with a child, family, or young person than it was in the past. I propose that to achieve healing in the environment of families that live what most people would consider to be very chaotic lives, some accommodation of chaos is necessary and its presence should not only be tolerated, as some sort of unwanted burden but welcomed, as it has an educational role for individual practitioners, the agency in general, and family members. After all, in a ‘good enough’ healthy family a certain amount of chaos is accommodated and is evident in many aspects, e.g. temper tantrums by children, unreasonable behaviour by teenagers, arguments between spouses/partners/siblings that take on a life of their own, various characteristics and idiosyncrasies of each member, unspoken emotional issues, accidents, different reactions to similar events, changes and growth over time. Indeed, a family wouldn’t be healthy without a tolerance of a certain amount of chaos! However the balance between ‘order’ and ‘chaos’ leans towards ‘order’ so that a healthy growth pattern prevails over time.
Good design in agencies that protect children in families affected by imprisonment not only accommodates and tolerates the chaos, unpredictability, and uncertainty that constitutes the reality that is encountered day-by-day, but takes cognisance of its existence and welcomes it, using it in a proactive manner as a tool for the growth, that is, as we have learned from fractal geometry, full of uncertainty anyway, that needs to occur and which is the principal aim of the work. Allied to this is the reality of the ‘feedback loop’ of self-similarity described above, utilised to bring about change, not to mention the educational value that it has for practitioners.
The result of the inclusion of chaos is that movement towards constructive and responsible patterns of behaviour is encouraged by self-similar, constantly repeated patterns of behaviour over time. These self-similar patterns are not, of course, exactly the same, in which case they would be boring and unattractive, but are experienced in different situations and environments at different times. Their power results from a mixture of their diversity and irregularity and is achieved by modelling and mirroring, resulting in a steady move towards patterns of responsibility which eventually reaches ‘Critical Mass’ in the individual or family. Modern developments in neuroscience are now proving, scientifically, that love, compassion, warmth, creativity, intuition, and other factors in our human experience that invite elements of chaos into our lives, have significant healing roles to play in our work. These beliefs have been known intuitively for centuries and person-centred therapy applies them in practice. While the pendulum has swung away from such methods in recent years, it is hoped that scientific affirmation will encourage a swing back in the forthcoming decades.
I believe that as humanistic psychotherapists we have a responsibility to preserve such elements, that is, the love, compassion, warmth, creativity, and intuition mentioned above applied over as long a time as is necessary, in particular, promoting them in initiatives aimed at people who, for a variety of reasons, struggle to be part of mainstream society and who probably need them most. While this article has focused on theories and concepts, the more practical ‘how to’ will be the subject of another article in Inside Out in the near future.
I found it interesting that when I began considering the role that fractal geometry had to offer in understanding unpredictability, uncertainty, chaos etc. in human behaviour, I was struck by the fact that the history of a family is often called a ‘family tree’, the life form that is very often chosen to demonstrate fractal geometry in the natural world!
Larry de Cléir MIAHIP MIACP, is Project Leader of Bedford Row Family Project, Limerick (www.bedfordrow.ie), an agency that supports families of prisoners and ex-prisoners. He has applied the concepts/theories in this article in the design of a Family Support and Crisis Intervention Course in communities affected by imprisonment. email@example.com
Lorenz, E.N. (1963): ‘Deterministic Nonperiodic Flow’ in Journal of the Atmospheric Sciences. 20: 2: 130–141.