
The use of mathematical techniques, theorems, and proofs in understanding geographical forms and relations. Two main types of application exist: statistical methods, which are employed in generating and testing hypotheses using empirical data, and pure mathematical modelling, which is employed when deriving formal models from a set of initial abstract assumptions. The two types come together in calibration: statistical methods are used to estimate, and test the significance of, various parameters associated with a given mathematical model, e.g. the friction of distance coefficient of the gravity model.
Statistical methods first began to be generally adopted by geographers in the 1950s (Burton, 1963). Consisting initially of descriptive statistics, there were also attempts at hypothesis testing, using, for example, chi square. Bivariate regression analysis was also tried, but it was not until the 1960s that the general linear model was fully explored (e.g. in factorial ecology). Since then much attention has been paid to a set of very sophisticated dynamic linear (e.g. spacetime forecasting models) and nonlinear (e.g. spectral analysis) statistical techniques, including those that bear peculiarly upon geographical problems (e.g. spatial autocorrelation).
More generally, since the first flush of enthusiasm back in the early 1960s, the history of statistics within geography has been one of trying to overcome the biased results that nonspatial statistics produce. Nonspatial statistical techniques, such as chi square and those based on the general linear model, produce bias when used on spatial data because of the effects of spatial autocorrelation; that is, sample points which are spatially proximate will tend to have similar values because of that proximity, and as a result will not be statistically independent of one another. After forty years of work, however, geographers have now developed a number of methods for correcting those biases, and for performing distributionfree tests on spatial data (Griffith, 1988; Haining, 1990).
Mathematical modelling. Inspiration for this second use of quantitative methods came most immediately from three sources:
{img src=show_image.php?name=2022.gif }Â social physics, which focused on spatial interaction among a set of discrete points and was represented initially by the gravity model, and later by the entropymaximizing model; {img src=show_image.php?name=2022.gif }Â neoclassical economics, which was concerned with optimization models around rational choice theory, and influencing geography principally through the regional science movement and location theory; and {img src=show_image.php?name=2022.gif }Â networks and graph theory, a branch of mathematics concerned with topological diagrams, and initially used to represent route networks in transport geography, but later utilized in some of geography\'s most abstract theorizing in central place theory.Most recently, a number of spatial theorists and modellers have turned to the analysis of complex systems and large data sets (see geocomputation). Stemming from longstanding criticisms made in human geography about the paucity of equilibrium frameworks, spatial complexity theory explicitly builds into its analysis nonlinear dynamics and interaction (cf. catastrophe theory; chaos). Such mathematical models allow for such useful properties as path dependence, shortterm unpredictability, and sensitivity to initial conditions and external perturbations (for more details and examples, see National Research Council, 1997, pp. 923).
In these various applications, the use of mathematics has been justified on a variety of grounds, but they amount to the same basic claim: that the world itself and the world of mathematics are fundamentally ordered according to the same arithmetic logic. As Galileo puts it, \'mathematics is nature\'s own language\' (Barnes, 1994).
But since the first heady days of the quantitative revolution, geographers have become increasingly leery of quantitative methods in particular and their broader intellectual justification in general. In the first backlash, both radical and humanistic geographers argued during the 1970s that many questions asked by human geographers could not be effectively answered using mathematics â€” as Harvey (1973, p. 128) famously put it: \'The quantitative revolution has run its course, and diminishing marginal returns are setting in.\' Such scepticism was bound up with a more general philosophical critique of positivism within human geography occurring at the same time. That argument in its most unsophisticated form was that because mathematics is the principal language of science, and because science is undergirded by the flawed philosophy of positivism, mathematics should be rejected. Such a position, however, established neither that geographers using quantitative methods ever practised positivism nor, even if they did, that positivism is a necessary justification for the use of mathematics.
More recently, from a poststructural perspective, a different kind of argument has been put forward that attacks the very logocentric ambitions of mathematics, that is, its belief in a fundamentally ordered world (Barnes, 1994; see deconstruction). Related also is work stemming from science and cultural studies that sees the use of mathematics, and statistical methods, as an effective vocabulary for social surveillance and control (Barnes, 1998; see also actornetwork theory; science, geography and). Mathematics and quantitative methods are not inherently ordered by some transcendental logic, but it can create order and discipline when applied to the social world.
Perhaps a more measured opinion of quantitative methods, then, is that like any specialized language they are very useful in answering some questions but not all questions (Pratt, 1989). Furthermore, the social entailments of the questions they do answer require continual scrutiny. (See also locational analysis; spatial analysis; spatial science.)Â (TJB)
References Barnes, T.J. 1994: Probable writing: Derrida, deconstruction and the quantitative revolution in human geography. Environment and Planning A 26: 102140.Â Barnes, T.J. 1998: A history of regression: actors, networks, numbers and machines. Environment and Planning A 30: 20323.Â Burton, I. 1963: The quantitative revolution and theoretical geography. Canadian Geographer 7: 15162.Â Griffith, D.A. 1988: Advanced spatial statistics: special topics in the exploration of quantitative spatial data series. Dordrecht: Kluwer.Â Haining, R. 1990: Spatial data analysis in the social and environmental sciences. Cambridge: Cambridge University Press.Â Harvey, D. 1973: Social justice and the city. London: Edward Arnold; National Research Council 1997: Rediscovering geography: new relevance for science and society. Washington, D.C.: National Academy Press.Â Pratt, G. 1989: Quantitative techniques and humanistichistorical materialist perspectives. In A. Kobayashi and S. MacKenzie, eds, Remaking human geography. Boston: Unwin Hyman, 10115.
Suggested Reading Burton (1963).Â Macmillan, B., ed., 1988: Remodelling geography. Oxford: Basil Blackwell. 
