The Lorenz attractor
Gleick, 1988, page 115
As with any history of any length, generalisations are made, details are skipped and complexities are ignored for the sake of narrative, in this section I must commit all these faux pas due to limitations of space, time and limited information; for this I apologise.
Science has always had high aspirations; from the work of Newton and his contemporaries a concept of the 'clockwork universe' emerged, with natural sciences seeking to be able to understand the elusive 'laws' that governed everything from the movement of celestial bodies to the interactions of subatomic particles. This ambitious paradigm reached its zenith in the works of the 18th century French mathematician and astronomer Pierre Simon de Laplace.
An intellect at any given moment knew all the forces that animate Nature and the mutual positions of the beings that compromise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movements of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain; and the future, just like the past could be present before its eyes
(Steward, 1997, p. 6, from Laplace's Philosophical essays on probabilities)
This awe inspiring passage, which is almost religious in nature, speaks of a world where everything is predestined: this would be a particular interest to archaeologists as "we could not merely predict the future but also retrodict the past" (Prigogine & Stengers, 1979, p. xii). This concept of predestination fits in with the ecclesiastical view that there was a divine plan and the vast intellect could easily be seen as an image of god. Scientists worked, and to some extent still work towards a total understanding of the world, Stephen Hawking is on a self-confessed quest for a 'theory of everything'.
People have problems with the lack of individuality that this paradigm give them. The church had always dealt with this by saying that we had free will, but the scientific community offered no parallel to this.
There once was a man who said 'damn!
It is bourn upon me I am
An engine that moves
In predestined grooves,
I'm not even a bus, I'm a tram!
(Maurice Evan-Hare, 1905 in Ferguson 1997)
This light-hearted example is a good manifestation of the discontent with the lack of choice that the above detailed mind set gave the individual. Certain social luminaries saw through the dogma and saw that things were not inevitable.
"there can be absolutely no inevitability" because "to imagine a human action subject only to the law of necessity, without any freedom, we must assume a knowledge of an infinite number of spatial conditions, an infinitely long period of time, and an infinite chain of causation."
(from the conclusion of Tolstoy's War and Peace, in Ferguson, 1997 page 36)
Whilst this mind set was extremely productive - it took us through the industrial revolution and on to most of the scientific wonders that make life in the modern world so wonderful - it is based upon a lie. Laplace was wrong. For reasons beyond the brief of this paper, from Heisenberg's uncertainly principle (Hawking, 1988), to the classic three body problem (Stewart, 1997), to Einstein's falsification of conventional concepts of time (Hawking, 1988), , we see that the 'clockwork universe' is a fallacy, and that Laplace's 'vast intellect' wouldn't be able to predict next Wednesday's lottery numbers, much less see far into the future.
So if science could not answer the questions they asked themselves about nature with a their traditional reductionist methodologies, what should they do to predict what the world's various systems would do in the future? There are fortunes to be made if you can predict stock values, and lives to be saved if you can predict hurricanes. As is usual in science, when a certain discipline hits a dead end, innovation comes from somewhere else. . .
Edward Lorenz, a meteorologist with a background in mathematics, constructed an abstract model of weather systems on a terribly primitive computer in the early 1960's. The program was based upon six variables and their relations, all went well until he needed to replicate a specific set of results.
When he re-entered the initial conditions, that had been rounded by two decimal places in the printout, at first the simulation looked similar to the original, but then it diverged, and within a short time bore no resemblance to it. The tiny differences in the initial conditions caused the outcomes to be entirely different. Lorenz concluded that 'In view of the inevitable inaccuracy and incompleteness of weather observation, precise, long range forecasting would seem to be non-existent' (Lorenz, 1963, p. 141) due to the inevitable roundings and inaccuracies inherent in the measuring of any real world system. This effect is very significant in itself, and has since been famously called the 'butterfly effect'. This has had a huge impact on its own, I shall deal with it at length below.
This could have been an entirely bad thing for science, but Lorenz saw beyond this. There were patterns to be seen, and Lorenz saw that a behaviour could be modelled, in which the properties of a system showed elements of periodicity and never strayed beyond certain values, but also never quite repeated a previous state. When plotted in phase space this draws the butterfly shaped 'Lorenz attractor'.
The Lorenz attractor
Gleick, 1988, page 115
The diagram has an imaginary third dimension, the lines never actually meet, but go behind and in front of one another, so the system never repeats its self.
Lorenz's concept was a strange beast - on a local level it was unpredictable, and even the most complete forecast would be useless beyond a few iterations, but on a global level it was stable (Gleick, 1988, p.48) - there is no way of knowing what the temperature will be this time tomorrow (except through waiting), but it will get warmer as summer approaches. We take the cycling seasons so much for granted that we forget that the inherent randomness in the weather follows a regular pattern.
These cycles and patterns are however not utterly stable, otherwise everything would remain in its allotted quasi-periodic cycles. There are points in these systems, whether they be due to a happen-stance configuration of influences, or the breakdown of another cycle, where the system can jump to a different cycle and behave differently. These points are called points of bifurcation. Practical examples of such points could be anything from the breaking up a smooth rolling fluid into a mess of turbulences to the French revolution or a cardiac arrest. But even within these bifurcations there is a form of order, when many of these are looked at together they can display their own chaotic behaviour.
These principles are all well and good, but further insight into the field is difficult without taking into account the work of Steve Smale, a topologist with an interest in dynamic systems. Smale's work (as referenced in Gleick, 1988) allows one to visualise the possible behaviours of a system in terms of phase space, he also provides purely mathematical model for unpredictability in the form of 'Smale's horseshoe'. I shall not deal with it in any great detail, but if you take a lump of topological matter and repeatedly fold it and stretch it then you will find that the distance between any two points in the original lump varies chaotically as the folding goes on (I am assured this is more impressive than it sounds).
His major contribution was however the description, in visual terms, of strange attractors (which, strangely, don't necessarily attract). There are four basic types of attractors, the sink, the source, the saddle and the limit cycle. These can combine in various ways to produce any number of possible repertoires of behaviour.
To visualise these attractors you must imagine the system at this instant as a ball bearing (bear with me) and the system's possible behaviours as a landscape (with gravity). If you place the ball bearing anywhere on the landscape, moving in any direction you like, at any speed you like, you can see how the system will perform under those conditions.
Visualise the attractors as follows:
The impressive thing about this method is that the behaviour of any dynamic system can be modelled, though there is no real way of knowing the shapes of the terrain until the system has run its course. This approach is particularly suited to archaeologists, who tend to have fairly modest mathematical skills, but excellent imaginations, it is much easier to imagine fluctuations of hunter gatherer forging patterns as a little ball bumping its way round a lumpy landscape than as a long string of figures.
The next aspect of chaos I shall deal with is the strange and beautiful world of fractals, made popular by the strange images in Mandelbrot's coffee table book (1983) and displayed in innumerable student rooms. People had seen what we now call fractals, such as the Koch curve and the paradoxically infinite length of our coast-line, and had dabbled with iterative mappings; if one looks hard enough there are also examples in literature.
So, nat'ralists observe, a flea/hath on him smaller fleas that prey,/and these have smaller fleas to bite 'em,/ and so proceed ad infinitum.
(Jonathan Swift, from Gleick, 1988, p.103)
Any progress beyond these first tentative steps was impossible without the mathematical brute force provided by the computer it to make out patterns in this, and so such things just remained mathematical curiosities, denied any serious enquiry.
It required a special combination of factors to take things any further; Benoit Mandelbrot was an insightful and imaginative mathematician, who had a peculiar gift for visualising problems. He was one of the first mathematicians to use powerful computers capable of portraying complex visual images, having access to an IBM research facility.
Mandelbrot first spotted fractal behaviour in the prices of cotton over a period of time approaching a century; economists had traditionally thought that the local, daily fluctuations in price were unrelated to the longer macro-economic trends, the former being due to random noise of individual activity, and the latter being due to changes in society at large, such as wars and recessions. Mandelbrot showed this to be nonsensical, as, if you looked at the fluctuations from any scale, they looked similar, there was a new type of symmetry present, now called self similarity, the 'roughness' of the curve was constant.
This attribute, of being rough, lumpy and irregular is much more redolent of nature than the Euclidean models with which people had been approximating. Britain is not a triangle, with conical mountains, and spherical clouds, and so fractal models of real world systems can be said to be more accurate representations than the classical Euclidean ones.
Mathematics needed a way of measuring this 'roughness', and Mandelbrot provided one.
First imagine a ball of fine string , look at it from a long way off and it appears as just a point, it has no length, area or volume, and so requires no numbers to describe it, it is a zero dimensional object. As you get closer to it, it looks like a ball, a three dimensional object, but when you look closely at the string it goes from appearing three dimensional to being one dimensional, having only length.This shifting in apparent dimension doesn't make any sense, but it is not unreasonable.
Another example is that of a ball of paper. Take a two dimensional sheet of paper and crumple it up into a ball: how many dimensions does it have/ exist in? Is it two or three, or somewhere in between? The 'somewhere in between' was nonsense before Mandelbrot, but he used fractional dimensions (this is not the root of the word 'fractal' which is from the Latin for 'to break' (Gleick, 1988)) to describe the lumpiness he had discovered, which makes sense given that a fractal depicted in two dimensions is an infinitely long one-dimensional curve constrained in a limited two-dimensional space (there are parallels for the phase diagrams of strange attractors, for example: the Lorenz attractor). The famous Koch curve is actually a 1.2616 dimensional object.
Of more immediate use to us as archaeologists, Mandelbrot applied the same techniques to the fluctuations of the Nile as he did to the cotton prices, and found a remarkably similar pattern, of the short term fluctuations looking remarkable similar to the systems long term behaviour. There is a very immediate and important piece of information for us as archaeologists that emerges from this principle. When we look at graphs of climates, whether they are of the temperature over the last week, or the last 100,000 years, we see a fractal.
As archaeologists we are used to treating the paleo-environment as a number of friendly predictable ice ages, but if one looks at the actual temperatures gleaned from deep-sea core analysis one will see that there really were no cycles or overall trends in temperature, so the breaking down of the past into glacial and interglacial is over-simplistic and myopic.
Self similarity, as I mentioned above, is a simple enough principle if one ignores the 'why'; it simply states that in a fractal system there are similarities in the small and the large scale. The roughness of the system is constant, so if one zooms in on a fractal, it will always behave in a similar manner. This is important due to the fact that it is very unusual; when a shape is renormalised (mathematical term for 'zooming in miles') one usually gets simplification. This concept is clearly present within society, motivations and interactions one can see between nations have an innate connection with the way that five year old children act with one another, and the analogies between these scales is a practical one as well as a semantic one, and gives us a constant language with which to talk about human action on all scales.
Another feature that is part of fractal mathematics only becomes clear if one watches a computer plotting an iterative map. At first the points scatter around the screen, with no discernible patterns, then as the picture becomes more detailed there appear to be lots of points in certain areas, fewer in others and none in others; these patterns are often utterly invisible if one were only to deal with unplotted reams of statistics.
If we have any two values we wish to compare then we can plot them in this way, and patterns can emerge allowing a greater understanding of what is going on, as spaces and clumps must have reasons (though these are usually obvious, no-one questions the validity of a distribution map if it has no mediaeval villages in the middle of the Atlantic).
Further to this, it may be possible for a computer to see patterns in these statistics that are not apparent to the human eye by using comparisons to plots from chaos. Indeed, any plot that does not represent a straight line has to be indicative of a non-linear system. There is the ever present problem, typical in archaeology, of lack of information, approaches such as this require vast amounts of information, and as such will not always be appropriate.
Gleick states:
Mandelbrot's work made a claim about the world, and that was that odd shapes carry meaning. The pits and tangles are more than blemishes distorting the classic shapes of Euclidean geometry. They are often the keys to the essence of the thing
(Gleick, 1988, p.94)
This interest in the details, rather than the misleading sweep of history, is encouraging to us as archaeologists as we are reminded that we do not have to work to the agenda of the historians, painting broad brushed pictures of the rise and fall of civilisation, but we can revel in the minutia.
Many have since climbed upon the shoulders of these great men, their stories may well go unrecorded, but their contribution to the discipline will make a lasting difference to all of our lives. I shall now deal with a few more principles of the discipline that are not included in the above history.
Shaw (in Gleick, 1988) formulated a novel approach, that could be seen as applicable to archaeology. He reacted against the scientific methods of examining a system by reducing it into equations governing its components to predict its behaviour; he ignored the entire system except the binary data that it produced. The specific case he was dealing with was a dripping tap, but this approach made the nature of the studied phenomenon irrelevant, it allowed him to ignore all the complications implicit in dealing with real world systems.
The most important thing that we can gain from this way of working is the ability, if we have a tame mathematician, to hand over our data for analysis, in the same way we do with things such as animal bones, and have a report returned to us, and all we need to do is be able to interpret what the patterns mean about the thing studied. This frees us from the need to be able to deal with the fearsome mathematics that chaos is constructed from, and hence makes chaotic archaeology more accessible to all of us.
My last point in this section is not so much a technique of chaos as an interesting aside proving that chaos is present in the universe at large, and that I am not just making tenuous connections.
The second law of thermodynamics, more commonly known as the principle of entropy, is an axiom of conventional science. Uvarov and Isaacs defines it as:
. . .the entropy of a system is a measure of its degree of disorder. The total entropy of any isolated system can never decrease in any change; it must either increase (an irreversible process) or remain constant. The total entropy of the universe is therefore increasing, tending towards a maximum, corresponding to complete disorder of the particles in it (assuming that it can be treated as an isolated system.
Uvarov and Isaacs (1986, page 136)
This holds true in all scientific models (this is why it is called a law), but if you apply it to a real world, involving living systems, it no longer makes sense. It is the nature of living things to organise things, be it the molecules that make us up, to organise stones to make our houses and so on. So out of the maelstrom that is supposed to ensue due to time we 'crystallise' into our organised bodies and set about structuring the world around us. The original case of order out of chaos. This begs questions of whether we will ever reach an equilibrium with the universe . . .
© Joe MacLeod-Iredale 1998