### 2. What is a mathematical model?

This begs the much larger question — what is mathematics? It is clearly much more than the popular image of arithmetic, more even than the schoolday memory of geometry, algebra and perhaps a little calculus. Poincaré said

'Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged' (Poincaré in Newman ed. 1956, 1558)

highlighting the all-pervading nature of mathematics and its dual role as 'Queen and Servant of the Sciences' (Bell 1952). Mathematics can be seen as a tool (or better, a tool kit) for the undertaking of research, but like all good tool(kit)s it comes with a set of manufacturer's instructions which users ignores at their peril.

Just as there are many branches of mathematics, so we must expect there to be many different types of mathematical model corresponding to them. One of the most fundamental aspects of mathematics is that known as set theory (Doran and Hodson 1975, 12). Although considered too simple to be useful in archaeology (ibid., 13), its extension, graph theory, is implicit in and foundational to the theory of Harris matrices (Harris 1979), the basis of modern excavation and recording techniques. A particular application of graph theory, known as space syntax (Hillier and Hanson 1984), has been found to be very useful in architecture and urban planning, and is beginning to find uses in archaeology. It is interesting that more advanced uses of set-theoretic models (e.g. by Soudsky 1973) have not captured the archaeological imagination in the way that the simple but illuminating Harris matrix has.

Geometry is another branch of non-numerical mathematics that has given rise to a family of models in archaeology. For example, Thiessen polygons are a simple geometric division of two-dimensional space that can be applied to the study of archaeological settlement patterns (Hodder and Orton 1976, 59-60). An extension of the Thiessen polygon approach, the XTENT model, was developed to take account of the different levels of site in a settlement hierarchy (Renfrew and Level 1979). The continued popularity of Thiessen polygons amongst archaeologists contrasts with the lack of uptake of the XTENT model — a situation which parallels that of the Harris matrix noted above.

The common archaeological perception of a mathematical model is perhaps better captured by the class of deterministic models, in which an algebraic equation is constructed to represent the situation of interest. Demographic models are a common example of this category (e.g. Hassan 1981); a further well-known example is the 'wave-front' model for the spread of early farming in Europe (Ammerman and Cavalli-Sforza 1973; 1979). Simple differential equations are a common feature of this class of model. A major problem of this class is the question of how their 'usefulness' can be assessed. No deterministic model can be expected to 'fit' the relevant data exactly (even if the model were correctly specified, errors in the data, inhomogeneous conditions, intractable boundary conditions, etc., would see to this), so we have to ask how big a divergence would leave such a model still 'useful'.

One answer is to build a random element into the equation, representing (for example) the effects of sampling on the recorded data values, or our own uncertainties about particular factors or relationships. This leads to the class of stochastic models, which have a range of outcomes rather than a single outcome, thus enabling models to be more realistically tested against data. A sub-class, which is growing in importance, is that of Bayesian models (Buck et al. 1996). In this approach, a result from the theory of conditional probability gives rise to an equation that (in principle) allows an archaeologist's 'prior belief' about a situation to be updated by data to create a 'posterior belief' based on both. The point to be made here is that it relies heavily on an explicit statistical model of the problem. It is beginning to make a serious impact on dating studies, through the use of the computer packages OXCAL (Bronk Ramsey 1995) and Bcal (Buck et al. 1999).

Probability theory leads to several families of stochastic models that may have applications in archaeology, for example Markov chains. The Markov chain model allows us to estimate the parameters of a process of change (e.g. in the condition of a collection of museum objects, see Orton 1996), from which predictions (e.g. about the future state of the collection) can be made.

Sometimes the equations of deterministic and stochastic models can be solved algebraically, but sometimes they cannot, and a numerical solution must be sought. This is usually done through computer simulation, using what is known as the 'Monte Carlo' approach, in which the distribution of a variable of interest is described by means of repeated simulation (Hammersley and Handscomb 1964). The behaviour of more complex models can also be examined through computer simulation (Sabloff 1981), and recent developments have shown how even simple models can yield surprisingly complex outcomes, which can only be achieved by computer simulation and cannot be predicted algebraically (Doran 1997; Kohler and Gumerman 2000).