The Utility of Mathematical Constructs in Building Archaeological Theory

Abstract

The utility of mathematical constructs in building archaeological theory Published in: Mathematics and Information Science in Archaeology: A Flexible Framework. A. Voorrips, ed. Studies in Modern Archaeology. Vol. 3. pp. 29-60, Helos: Bonn (1990) THE UTILITY OF MATHEMATICAL CONSTRUCTS IN BUILDING ARCHAEOLOGICAL THEORY Dwight w. Read INTRODUCTION 1 In his book, Analytical Archaeology, Clarke (1968:512-513) noted three ways in which mathematical concepts are relevant to what he calls archaeological ideology: (i) The need for entitation and quantification ... using ... descriptive statistics ; (ii) The need to handle relationship concepts ... using analytical inductive statistics ; (iii) The need to handle the regularities in complex data in terms of isomorphic systems of symbols arranged in axiomatic schemes, models, or calculi A similar theme was iterated by Cowgill (1986:369) in a review article titled. Archaeological Applications of Mathematical and Formal Methods. There he referred to three broad categories comprised of archaeological observations, analytical methods, and sociocultural theor , but then observed that although some theory is expressed directly in mathematical terms ... the vast majority of archaeological uses of mathematical and formal techniques pertain to the domain of analytical methods or to the design of data collection . And in a recent text, Anthropological Archaeology, Gibbon (1984:383), though espousing the value of formal and axiomatically expressed theory in archaeological reasoning, bluntly commented that No theory within archaeology has ever been formalized . Diametrically opposed conclusions can be drawn from these comments: (1) the lack of formalized theory is indicative of the immaturity of archaeology as a science, or (2) formalized theory is largely irrelevant to the development of archaeological theory. The intent in this review is to show that lack of substantive, axiomatic-like theories in archaeology is neither inherent to the discipline, nor to the capabilities of the discipline's practitioners, nor to the alleged irrelevancy of axiomatically framed arguments for an archaeologically based theory (Salmon 1982). Rather, mathematically based techniques in the form of statistical methods and modeling have already been well-established as an essential part of archaeological data analysis (Read 1989). What is lacking, though, is application of mathematical formalism to the theoretical issues of archaeology, despite recognition of the value of axiomatically or formally expressed theory as shown in the above quotes. I suggest that the disparity between (1) the acceptance of statistical methods and (2) the lack of application of mathematical formalism stems from inadequate understanding of the way mathematics provides not only a language for the expression of relationships, but also a means for reasoning about their consequences, hence a language for extending archaeological reasoning. To develop this argument, I will first consider the nature of mathematics and its several roles when serving as a language and conceptual system for expressing relationships and processes responsible for the structure found in data. Then I will examine several published applications of mathematical formalism directed towards the understanding of processes. The first topic will make a fundamental distinction between mathematical methods used to express idealized patterns surmised from data and mathematical formalism used to

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