Graph constructed from the populations of US urban districts
Dendrinos & Mullally,1981
Extensive research has turned up remarkably few applications of chaos theory to archaeology and to the social sciences, whereas there are many uses of chaos theory outside the social sciences. This can, to a large extent, appear to be due to the mathematical ability required being rare within the social sciences, and also due to the hermeneutic approach currently in vogue with archaeology and its related disciplines.
As there is so little literature dealing with the subject of this paper, I shall widen the scope of my review to deal with the few cases in fields related to archaeology, those from which we draw information and inspiration, and those in which we can see the possibility of valuable analogies.
I shall deal with the texts in chronological order, in order that trends may be seen. At the end of this section I shall deal with these trends.
I have already dealt with this influential article in the 'History of chaos theory' section above; this is the article that set the whole paradigm shift in motion. Obviously it does not say anything specifically archaeological, being a paper concerning meteorology. He does however deal with some issues that affect both meteorology and the social sciences, both of which dealwith immeasurably complex and vast systems. Because of this there are many quotes, which, when taken out of context, can lead to insights as to why the processual movement has seemingly failed in its aim of systematising archaeology.
A non-periodic solution with a transient component is sometime stable, but in this case its stability is one of its transient properties
(Lorenz, 1963, p. 141)
Here we see the first inkling that the paradigm I have coined 'Laplacean' is doomed, since stable systems only appear so on a short term basis, and so it is impossible to glean any information about the behaviour of a long term system (system in the case of real world examples meaning the whole universe, because, as is stated above, every particle in the universe influences every other). This idea can be applied archaeologically by debunking the concepts of stable ecologies with human hunter-gatherers as a component part. The systems that the processualists constructed may have worked for the available data, but if their view of reality were the true one then we would never have been able to escape from their cycles and linear relations. There is without doubt a lot to be usefully gained from a processual approach to the archaeological evidence (look where it took science ), but the practitioners must realise that they are only dealing with models, the integrity of which is shaky at best.
Lorenz also realised that the weather would only ever act in an entirely predictable way if two analogous states could be found. This idea, of two identical states, is patently ridiculous if applied to culture, though the quasi-cyclic behaviour of our ancestors has often been treated that way.
In this short article Gilpin identifies the presence of a strange attractor in the numbers of prey in an ecological system, and uses this to ask how chaotic ecological systems are, and hence how wrong existing models are.
The message from this is that even the simplest possible models of community interaction require exhaustive cybernetic analysis before their repertoire of behaviours can be known, and some of these are likely to be complex
(Gilpin, 1979, p.308)
In this extract he hints at a 'repertoire of behaviours' of a community of creatures, so he is not even attempting to predict the actions of the complex system with which he is dealing; he is using the strange attractor to discover the range of behaviours present in the system. Archaeologists have always done this in their quest to understand what people were doing. They predict the activities of the ancestors, whereas surely it would be more rigorous to keep one's imagination in check and 'retrodict' the range of possible activities of the group, rather than intuitively and arbitrarily choosing one of the possibilities.
The degree to which real ecosystem behaviour is chaotic is possibly the most fundamental question facing community ecology
(Gilpin, 1979, p. 308)
This is an interesting point, especially as there is a distinct overlap between the field in which Gilpin is working, and the direction from which I am coming. The systems we are dealing with are undeniably chaotic in the classical sense of the word, but to what extent are they chaotic in the new, theorisable way? It is a difficult thing to find; the variables in human society are often entirely arbitrary, and to take into account even the most important relationships in a system can require a staggering number of dimensions to be considered. Because of this, finding chaotic relations in human society will be many orders of magnitude harder than those found in Gilpin's simple relations between predator and prey.
This book is similar to Casti in that it has the feeling of being an idea that obviously had some merit, and so concepts and ideas have been collected to support the idea behind the work. At the time of writing this article chaos was only known to a few in the high sciences, well before the popularising influence of Gleik's 'Chaos' (1988) and Stewart's 'Does god play dice?' (1997) propagated the message of the new science, and as such it doesn't deal with chaos as such, but their work is none the less useful. To summarise their operational basis (as outlined on page xv): They hold that while some parts of the universe may operate like machines; these are closed systems, and closed systems at best form only a small part of the physical universe. Most phenomena are, in fact, open, exchanging energy or matter (and, one might add, information) with their environment. Surely biological systems are open, indicating that attempts to understand them in mechanistic terms are doomed to failure. They use such phrases as 'ceteris paribus'- all other things being equal (Prigogine & Stengers, 1979, p. xi), which makes no sense at all to the ears of a chaotician. These considerations aside, this work does have quite a lot to offer those of an archaeological bent.
There exists. . . a model, defining the optimal spatial distribution of centres of economic activity. Important centres would be at the intersection of a hexagonal network, each being at the intersection of a hexagonal network, each being surrounded by a ring of towns of the next smallest size
(Prigogine & Stengers, 1979, p. 197)
This is patently referring to a fractal, and it is more than likely that archaeologists have used this model to deal with past settlement patterns. The fractal nature of this phenomenon allows us to see rather deeper into its structure than is possible without such an understanding; for instance, it is more than likely that if you combine this work with that of Dendrinos and Sonis (1990) then it would be possible to construct a picture of the underlying behaviours of interacting urban units. It is also a real possibility that such an approach might shed light upon the first aggregations of humanity into settled urban systems (Lord Renfrew's work springs to mind)
In the summing up to the first section, they come to the following interesting conclusion (the following is a paraphrase as the original contained a lot of jargon specific to this book):
Not only are biological systems open, they have to be open. . . they feed on the flux of matter and energy coming to them from the outside world. Any ecosystem isolated from the universe as a whole will die. This explains why the exact patterns they obey can never be predicted using conventional methodologies.
(Prigogine & Stengers, 1979, p.127)
This concept seems muddled and a little odd to one with a good understanding of the principles of chaos, but its conclusions are roughly the same as those given by a complex system, they both deal with systems that have an unpredictable element.
The article also deals with the interaction between science and culture at large, and has a interesting and seemingly reasonable view of science, seeing it as being part of society, integrated with the whole by multifarious feedback loops (both positive and negative) (Prigogine & Stengers, 1979, p.xii). This view of science shows it as nothing more than an aspect of society, and so shows the 'holy grail' of scientific truth to be nothing more than a culturally imposed phenomenon. They follow this up with this passage accredited to Herman Weyl:
Scientists would be wrong to ignore the fact that theoretical construction is not the only approach to the phenomenon of life; another way, that of understanding from within (interpretation), is open to us. . . Of myself, of my own acts of perception, thought, volition, feeling and doing, I have a direct knowledge entirely different from the theoretical knowledge that represents the parallel cerebral process of symbols. This inner awareness of myself is the basis for the understanding of my fellow men whom I meet and acknowledge as beings of my own kind, with whom I communicate with so intimately as to share joy and sorrow with them.
(Prigogine & Stengers, 1979, p.311)
This approach, whilst familiar to us as being a typical sentiment of the post-processual movement in our own field is unusual at this date and in this field. Such sentiments of disaffection with the 'white coated' scientific truth, whilst still feeling that there was more to the universe than the purely experiential is common amongst those, such as myself and the authors of this piece, who see chaos as a valuable tool to extricate our selves from the morass of contradictions that such a view holds.
Firstly, it must be stated that the maths in this article is very penetrable, it is a refreshing change from those, such as Casti (1983) and Prigogine and Nicolis (1997) who belie the interdisciplinary nature of what they are attempting by requiring a mathematical knowledge above and beyond that of practically anyone outside the 'hard' sciences such as astronomy and engineering. This approach restricts the use of the insights contained in the articles to those who work in related fields, another example of the self-reinforcing pigeon-holing of academia.
This is one of the very few articles there are that are directly relevant to the archaeological community, it deals with the fluctuations in the populations of various American urban districts. The city is perhaps one of the single most complex social phenomena in existence, and as such it is impervious to analysis by any form of reductionist methodology, and as such we must approach the subject from a different direction. Dendrinos (an urban planning lecturer) and Mullally (an economics professor) choose population (logically) as their quantity to study, and make use of the extensive census records to plot the behaviour of 100 urban areas in the US through time on phase diagrams. The trends and cycles identified are plotted on the graph below.
Graph constructed from the populations of US urban districts
Dendrinos & Mullally,1981
We see here that the cities do indeed conform to specific categories of population change. It is interesting that there are clear strange attractors and steady states present in utterly complex systems, systems that contain us (and I for one don't think of myself as terribly predictable).
This information could be very useful to archaeology; it goes without saying that archaeologists spend a lot of time analysing the remains of cities, and I see no reason why a little uniformitarianism would be amiss here. From this we can see that populations of urban centres follow trends other than the traditional choice between steady state and the exponential form of growth. The only significant difference between the modern examples and those studied by our fellow archaeologists is the higher rate of flux in the modern cities due to the unprecedented mobility of their populations.
To summarise the findings of the article, the cities studied tend to act as a damped oscillator around a set value, once they reached this they remained in a steady state. Their drift can be upset at any time if the system is perturbed, and when perturbed it reaches a finds a new focus of its oscillation. The perturbations could be due to large population upheavals (e.g. the explosion in car production in Detroit in the first half of this century): these factors are beyond the remit of the essay, but provide an intriguing avenue of enquiry.
As I mentioned above, some of the papers are so impenetrable as to make them practically useless to non-mathematicians. I include a sample of the kind of text that typifies this, and many other articles.
Theorem 2.
(a) suppose that the zero solution to equation (*) is stable (asymptotically stable) (unstable). Then the zero solution of equation () is also stable (asymptotically stable) (unstable).
(b) Suppose that the zero solution of the equation (*), let x(t), y(t) be a solution to the equation (
This consideration apart (which meant only the introduction and conclusion are dealt with here), Casti tries to apply 'simplexity' (a term coined by Stewart) to a variety of different systems. He appears to be looking for a question for his answer, which is an unreliable approach at best, and in his work I can't see anything revelatory in the work.
In this work Mayntz appears to be working from a similar type of inspiration to myself. She asks 'Can the ideas behind chaos be applied outside the natural sciences?'. She uses this mind set to approach a commonly held view amongst multifarious groups from ecologists to statesmen that our society is an innately fragile one, and without strict control and shepherding it is likely to collapse into a self-centred and destructive anarchy.
The paper approaches some of the major cataclysmic events that have befallen Western society in the past few hundred years, and the way in which historians tend to trace the causes of these to relatively insignificant causes: did the baker in Puddle lane really cause the great fire of London? Did the assassination of Archduke Francis Ferdinand (Infopedia, 1995) really spark off the First World War? Surely something requires a multitude of factors to be conducive to instability before any such upheaval could occur, or are we looking at the butterfly effect on a grand scale?
A beautiful example of the butterfly effect is given (page 12), although strictly speaking we can't tell if it is the butterfly effect as we can't rerun events to find out if things would be different given a small perturbation. A chain of reason is constructed from a small change within a massive institution to the collapse of the entire institution. For example, the abolition of the slave barracks led to the disappearance of the slave class and the emergence of a class of serfs bonded to the land, which in turn led to a shift from wealth being an urban feature, to its concentration in rural farmsteads, so leading to a population tied to the land, and hence the lack of a standing army and civil service, and hence 'The decline and fall of the Roman empire'!
She is, however, commendably wary about ill thought out interdisciplinary approaches, and speaks of being 'seduced by verbal analogies' (Mayntz, 1989, p. 4) of which non-scientists are often guilty. Newton's First Law 'For every action there is an equal and opposite reaction' has been co-opted by innumerable other fields, for which it has no basis in truth whatsoever.
There is a distinct feeling in this paper, that, although not explicitly stated, she believes populations have a distinct tendency to work towards a steady state, but when they achieve it they become discontented and work away from the stable state. This tendency can be observed in the way the status quo is most valued when society is unstable and in flux, and the dissident factions of society are at their most prevalent when the social situation is at its most stagnant. This could be classified in the language of chaos, as a damped oscillator, that, upon reaching a certain critical value, will self-perturb and find a new focus of oscillation. The speed of this turnover could be put down to the speed at which information is transferred around the group.
Also expressed are problems with the notion of distinct historic ( and prehistoric) eras, a problem which we in archaeology know only too well. This mind set requires, at least as far as historical resolution can take us, instant changes from one social system to another. We spend a great deal of time trying to pigeon hole sites, objects and behaviours into entirely arbitrary groupings, whereas social change is usually incremental.
All in all, this article is a perfect example of why archaeologists find it so easy to use concepts and approaches from sociology productively; after all we are studying the same thing, just in different tenses.
This short article takes a very peculiar stance, it appears to embrace chaos theory whilst explaining this through with rather contradictory reductionist methodology. He seems to draw a parallel between the aggregation of humans into power structures ( person › family › tribe › state › nation etc.) to the microscopic aggregation of elements into the building blocks of life (atoms › molecules › macro molecules › DNA) (Binning, 1989, p.32). I don't see how the spotting of small, probably coincidental, similarities does academia any good.
This paper springs from the context of a background of working with multi-scalar analyses. Archaeologists have traditionally encountered such principles through their contact with the Annales school, with their long, medium, and short duration views of the past. The multi-scalar analysis allows one to extract the tell-tale signs of quasi-periodicity, though one is made well aware of the fact that if the wrong time scale is used then the trends and cycles present may not show themselves, because if the time scale dealt with is too short then the patterns will not yet be visible and hidden in localised noise, and if the time scales are too long then the results will suffer due to loss of information. Hence in order that this does not become problematic one must be very aware that the time scale one is looking at is of paramount importance.
She also deals with the shortcomings of a scientific mind set when applied outside their traditional home amongst the natural sciences, she states that: 'Processual archaeology has the Newtonian goal of finding atemporal universal laws and theoretical models to describe and explain observed cultural patterns and processes' (Spencer-Wood, 1993, p.2), but that: 'anthropological models of culture neglect the possible large scale significance of individual variation' (Spencer-Wood, 1993, p. 3) and also due to the fact that regressions to linear relations are impossible in any 'macro' system, because of these factors the scientific approach falls down when applied to cultures.
She mentions an intriguing example of a chaotic distribution pattern, that of rumours. Would it not be possible to glean valuable insights into other information propagation systems, such as the spread of technology or belief by the study of the spreading of rumours which is comparatively easy compared to the practical impossibilities of studying the patterns of knowledge propagation in the past?
Her conclusion is summed up as follows: 'Research has led me to challenge the theoretical assumptions that 1/ order is separate from disorder, and 2/ only orderly patterns deserve study' (Spencer-Wood, 1993, p.4). In saying this she recognises the value of what chaos has to give to fields, such as ours, that are caught between the complexities of our evidences and the simplicities of our models.
This paper is very similar to a number of the others that I have encountered (including, to a certain extent, this dissertation). Olsen shares with many archaeologists the perception and imagination to realise that there are insights to be gained from an interdisciplinary approach; in this case drawing from the natural sciences, but in common with the vast majority of us, does not have the mathematical ability to cope with the actual mechanics behind the theories. This is a problem not easily addressed unless archaeology departments start recruiting maths graduates.
As in many of the other papers I have dealt with, one of the major motivating factors in the writing of the articles was a real disaffection with the processual, and anti-processual paradigms dominant in archaeology, the following excerpt says as much:
In the past, archaeologists have attempted to discover general laws pertaining to human behaviour in the archaeological record. They have also attempted to design simple predictive models to better understand the behavioural process. On the whole these methods have met with little practical success
(Olsen, 1996, p.11)
She also deals with the limitations inherent in our field. The resolution of archaeological and paleo-enviromental data is very poor, and as such we are denied the usual scientific boon of being able to pick the information we can deal with, we are restricted to the scant evidence we can glean from the ground. She also states the valuable point that most scientific analyses rely on initial conditions being known and this is not a possibility in archaeology because 'reconstructing initial conditions in a moment in prehistory is overwhelming and forever beyond our grasp' (Olsen, 1996, p.3). I would argue that there exist no initial conditions in the real world as everything has a past, and so there will always be a 'before'.
To conclude the more practical aspect of this paper she writes: 'strange attractor governs the migration patterns of prey animals, and so the movements of the hunters is that of a subsidiary strange attractor' (Olsen, 1996), which has the novel quality of being of obvious practical use to the archaeologist, and also making perfect sense to the chaotician (or chaologist).
This is a very curious book; as stated in the title, Ferguson deals with counterfactual and alternative histories. Many historians have expressed the opinion that this course of reasoning is pointless, things happened the way they understand them, but I would argue that such a conservative approach treats the past as though it were set, and that the history books 'are' history, rather than only someone interpreting the evidence. We have never really had this problem in archaeology, due to a combination of the youthful vigour of the discipline and the sheer lack of evidence.
It is always interesting to ask 'what if?' of events past, whether it is a personal 'what if?', or a historical 'what if?'. In doing this we are questioning the idea of predestination, and admitting the sensitivity our world has to its initial conditions, and what a contrived set of coincidences it is that has built the world we now live in. Ferguson also points out that history was once, and always will be, yet to come. It is easy to see the people of the past as living in the past, and as just fulfilling their roles to become us, but their 'now' was just as valid as ours.
The approach to the past as not set gives one a greater freedom to construct one's own truths about the past, and also gives us a greater confidence to look into the future, admitting that we will probably be wrong. It is also an entertaining exercise in imagination.
Ferguson does mention chaos theory in passing, pointing out that there are parallels between a strictly scientific understanding of chaos and the random path history takes, but he doesn't take it very far. The largest section with any connection with chaos theory is when he mentions that Darwinian evolution is based on the tiny reproductive advantage randomly bestowed upon an organism by the arbitrary mutation of its genetic makeup (Ferguson, 1997, p.75). This is a perfect example of the butterfly effect in action.
To conclude, Ferguson deals competently with the emergence of the intellectual world from its deterministic routes from a different angle to that I have become used to.
This book comprises of a section by the editors, McGlade and Van der Leeuw, and a number of articles from others who fit the type of a pluralist, non-linear approach to archaeology. The articles are very site specific and technical, and in many cases seem a little too abstract to be of any huge practical use, but one can hardly expect a new paradigm to emerge fully formed from minds of its instigators, as the authors themselves admit, 'Our exploration into non-linear properties resident in societal systems is a first and tentative one' (McGlade and Van der Leeuw, 1997, p. 3).
The approach this book takes is basically a pluralistic one. The authors are dissatisfied with the petty infighting that has split archaeologists into different camps who constantly gripe at one another and refuse to see the advantages in the other people's methodologies. The authors suggest that we should put our differences aside and 'steer a course between the oversimplifications inherent in so-called 'laws of human behaviour' and the limitations of extreme textualism' (McGlade and Van der Leeuw, 1997, p.2). They go on to say that each of the existing theoretical approaches has tried to address the issue of complexity but have been guilty of 'essentially obscuring the intrinsic complexity of the problem by superimposing a single theoretical lens through which the data are to be viewed' (McGlade and Van der Leeuw, 1997, p. 3). This is a refreshing and positive message for archaeology.
They also point out a mind set which most archaeologists unknowingly subscribe to. We have traditionally viewed the past as long continuities, followed by huge upheavals, we have the 'Roman Era' followed by the 'Dark Ages', the 'Bronze Age' followed by the 'Iron Age', these convenient little pigeon holes hide huge amounts of complexity amongst the hundreds of years and millions of people involved, and all because of a passing similarity in the assemblages from similar sites in time and space. This brings to mind parallels with the work of Mandelbrot; the realisation that a system that was once modelled as a smooth homogeneity was in fact 'lumpy' and rough, and surely it is especially in this arena of study, that most can be gained from the small details. It is empty, hollow information to know how many thousand years the 'Neolithic' was, but to reconstruct how it would feel to live in Çatal Hüyük is immensely rewarding.
They also deal with many other issues involved in an interdisciplinary, pluralistic approach to archaeology competently, but I have already dealt with them in other areas of this paper, and will avoid repetition.
To conclude: this book is the first work to bring a high level of understanding of the principles behind chaos theory to archaeology. All the other interdisciplinary works I have dealt with so far have been found wanting for either understanding of the deep science, or being answers looking for questions. I hope that this is an example of things to come, if this is the case then we may be looking at an impending paradigm shift for those of us with processual leanings.
I apologise for the brusqueness with which I am about to deal with this paper. I see no reason for any of her conclusions: if she understood chaos then she would see that complexity does not have to be, and indeed almost never is planned: there are a myriad of reasons why the Minoans built their structures as such, and some precognisance of non-linear dynamics is one of the least likely of them.
When looked at as a whole, the literature I have dealt with moves from the specifically scientific to a wider set of applications. This shows a (chaotic) diffusion of the principles from its origin in meteorology and topology to other areas of where it can be productively used. These applications are as yet not mature and have a number of problems (see 5. Problematisation). My paper is very much part of the vanguard. McGlade and Van der Leeuw's (1997) book could be the beginning of chaos' appropriation by serious archaeology, only time will tell.
© Joe MacLeod-Iredale 1998