Early attempts to model archaeological thought processes can be found from the end of the 19th century onwards, although whether the authors would recognise their models as such is open to question. Johnson (1972, 315-16) cites Boas' use of a similarity matrix (Boas 1895) and Czekanowski's work on seriation and clustering (Czekanowski 1911) as falling into this category. The example of Petrie's invention of seriation (Petrie 1899) has already been mentioned.
Despite this early start, the explicit use and widespread discussion of models appears to be a product of the 'New Archaeology' of the 1960s and 1970s. The stall is set out most clearly by Clarke, in a preliminary fashion (1968, 32-40) and later in more detail (1972). Here he makes a strong case for archaeologists to concern themselves about models (ibid., 3), and goes on to distinguish between 'controlling' and 'operational' models (ibid., 5) (see 3 above), and lists several types of mathematical models (ibid., 18-24) (see 2 above). In his enthusiasm he tends to broaden the term 'model' to include what might better be called 'hypotheses', but he nevertheless sets the scene for a decade of explicit model-building and -using.
Belief in the value of the explicit use of models was strong throughout the 1970s. In Renfrew (1973) we can see a wide variety of approaches and problems all coming together under the umbrella of 'models', a clear sign of a bandwagon effect. Towards the end of the decade, we can see more focussed and developed uses of models. For example, the contributors to Earle and Christenson (1980) make use of strategy cost curves, linear programming, decision theory, and least-cost models, in their studies of subsistence economies. The 'high point' of this trend can perhaps best be seen in Renfrew and Cooke (1979), in which, in addition to all these approaches, we are also presented with Catastrophe Theory (Poston 1979; Poston and Renfrew 1979), and 'perceptrons' (Reynolds and Ziegler 1979, 406).
After this, there seems to be a rapid decline in the quantity and sophistication of 'modelling' papers. The direction is indicated by Sheridan and Bailey (1981a), whose sub-title, 'Towards an Integration of Ecological and Social Approaches', shows that they are trying to reconcile 'economic' or 'ecological' approaches with more fluid 'social' approaches. In their introduction (1981b), they note that palaeo-economics was a dominant theme in British archaeology in the 1970s, culminating with Higgs (1975), and suggest that a more balanced approach is needed. It is significant that there is a marked lack of formal or explicit models throughout the entire volume, showing that the pendulum was already swinging away from their use. How far it swung is demonstrated by the fact that in a recent encyclopaedia of archaeology (Ellis 2000) there is neither a section nor an index entry for 'model' or 'mathematical' (they are there in the text, but one really has to search for them).
The turning point in the fortunes of mathematical modelling in archaeology that followed Renfrew and Cooke (1979) is worth examining in more detail. It could be seen as a working out in practice of Catastrophe Theory expounded by Poston (1979), with the whole approach being caught on its own cusp catastrophe. This is of course just a rather fanciful description, and does not really explain why it happened just then, or indeed why it happened at all (which may itself point up weaknesses in Catastrophe Theory). Various reasons can be suggested:
a. the use of mathematical models had been tested and found wanting. If even the highly sophisticated models used in Renfrew and Cooke (1979) do not lead to useful archaeological insights, is it worth bothering with mathematical models at all?
b. the level of complexity of mathematical modelling passed a threshold beyond which archaeologists were not prepared to go. An approach must develop if it is to survive, and if further development is perceived as too difficult, it may quite suddenly run out of steam. This situation can be paralleled by the failures to develop significantly beyond the use of Thiessen polygons and the Harris matrix, as noted above.
c. the mathematical modelling approach no longer answered the sorts of questions that archaeologists were asking. The trend from the urge to generalise (apparent in the New Archaeology of the '60s and '70s) to the emphasis on the particular and the individual (a feature of the post-processual approaches of the '80s and '90s) may have left mathematical models as a solution without a problem.
d. mathematical models may not have been discarded so much as 'gone underground'. In other words, they had become so much 'part of the furniture' that there was no longer any merit in writing about them explicitly. This is a common pattern as techniques pass from being 'cutting edge' to being 'mainstream' (for example, think of desktop publishing or computer-aided design).
In the context of simulation modelling, Aldenderfer (1981) saw the main reason as being (a), while Hodder (1982) preferred (c) as an explanation. There is probably an element of each of (a) to (d) in the overall picture, although the relationships between them raise interesting questions. For example, did the increasing complexity and perceived failure of mathematical models help to provoke the backlash that became post-processual archaeology, or would it have happened anyway? Alternatively, can post-processual archaeology be seen as an easy option in the face of a growing and unacceptable level of complexity?
Whatever the reason for this turnaround, mathematical models have nevertheless been making a small but significant recovery at the 'strategic' level in recent years, while 'tactical' uses have continued quietly throughout, as in reason (d) above. Lake (2001, 727) sees this revival as having at least three causes:
the narrowing focus has increased the likelihood of producing output utility,
the borrowing of principles from the biological sciences, especially behavioural ecology, has 'eased the process of validation',
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