Every society has at least one means of representing number - the set of (spoken and/or written) lexical numerals associated with a language. Additionally, some societies use one or more numerical notation systems - visual, non-phonetic systems of signs for representing number. Numerical notation is trans-linguistic - its signs are graphic representations that are not tied intrinsically to any one language - whereas lexical numerals are language-dependent. I will compare and contrast systematically the use of lexical numerals (one, two, three...) and numerical notation (1, 2, 3...) using a comparative anthropological framework. This interdisciplinary project will contribute to research in cognitive psychology, linguistics, and the history of science, as well as anthropology.

In my doctoral dissertation, The Comparative History of Numerical Notation, I examined all numerical notation systems ever used worldwide (over 100 distinct systems in all). I outlined the historical trajectories through which numerical notation systems developed, and analysed the structural properties of these systems and the extent to which they exhibit cross-cultural regularities. I examined both historical and cognitive-functional factors concerning numeration, finding that the lexical numerals of the users of numerical notation systems are one of these constraining factors. By augmenting data from my dissertation with new data on linguistic numerals, I can analyze more fully how humans represent and use number.

Because no one discipline lays claim to the social study of numerals, many interesting questions have been inadequately addressed. Recent important contributions on numerals have been made in linguistics (Hurford 1987), cognitive psychology (Dehaene 1997), and anthropology (Crump 1990). My research complements these contributions using two data sets that are only accessible through anthropological and historical investigation: numerical notation systems (already collected in my doctoral research) and the lexical numerals used in conjunction with them. Using documentary, ethnographic and archaeological evidence, as well as secondary literature, I would ascertain the structure of these systems, changes in them through time, and the contexts of their use. Where available, electronic text corpora for several languages will augment this data, and permit detailed analyses of techniques for numerical representation. The necessary data concerning lexical numerals are lacking for some societies; between twenty and thirty cases, chosen judiciously from those for which data is available, will suffice. By doing so, I hope to answer five important theoretical questions.

1) What are the cognitive effects of visual and non-visual representations of number? Using a comparative perspective, I hope to establish to what extent numeration is mediated by language and to what extent it is purely visual. Cognitive studies have examined the subject from both perspectives, but have rarely compared the two. Cognitive psychologists interested in the capacity to process visual information (Miller 1956) and the number concepts of infants (Wynn 1990, 1992) of necessity have examined visual rather than linguistic numeration. These visual approaches relate directly to numerical notation. A second body of research examines the cognitive effects of lexical numerals, investigating the learning of regular and irregular lexical numerals among preschoolers (Miller et al. 1995) or the frequencies with which different lexical numerals are used in various languages (Dehaene and Mehler 1992). While some experiments compare the use of lexical and graphic numerals among individual populations (Chincotta and Underwood 1997), cross-cultural comparison of the two is lacking. By comparing the use of lexical numerals and numerical notation in their contexts of actual use, I can use historical and ethnographic data to confirm or refute the findings of cognitive psychologists.

2) What similarities and differences exist in how numbers are expressed in language and in graphic symbols? Joseph Greenberg (1978) wrote a pioneering paper on universals of numeration describing 54 generalizations concerning lexical numeral systems, which he clearly believed to be mainly grammatical in origin. The belief that human numeration ought to be explained in linguistic terms is even clearer among generative grammarians (Chomsky 1988: 167-169; Hurford 1987). In my dissertation, I described around 40 generalizations that apply to numerical notation systems. There is remarkably little overlap between my set of generalizations and Greenberg's. For instance, while 300 is expressed in Roman numerals as CCC, no language expresses this number as "hundred hundred hundred". This begs for investigation, given that both techniques of representation apply to a single cognitive domain (number). It suggests that the medium through which number is expressed strongly influences the way in which numbers are conceptualized. If so, then by analyzing the range of variation in both lexical numerals and numerical notation, I can help theorize the cognitive importance of different expressive media.

3) What influence do lexical numerals and numerical notation systems exert on each other among users of both representational techniques? Because my data are anthropological and historical, I will study this subject contextually, rather than relying on experiment or conjecture. For numerical notation systems that developed without influence from other systems, I will see what influence the lexical numerals of the region have on their structure. For numerical notation systems based on earlier systems, I will examine the interaction between the lexical numerals of both the transmitting and adopting societies. The trans-linguistic nature of numerical notation probably makes it more conducive to intersocietal transmission, whereas lexical numerals traverse linguistic boundaries more rarely. Of course, a range of other social, political, and cultural factors also influence the transmission of both lexical numerals and numerical notation.

4) What relation exists between forms of numerical representation and social complexity? Greenberg (1978), as a linguistic anthropologist, applied a cultural-evolutionary framework to lexical numerals, and hypothesized a direct correlation between social complexity and the maximum expressible number in a language. Because the invention of numerical notation correlates with the emergence of early civilizations, an evolutionary perspective can help theorize its origin. Anthropologists have contributed extensively to this literature, but rarely systematically and never distinguishing adequately between lexical numerals and numerical notation (Steensberg 1989, Crump 1990, Schmandt-Besserat 1992, Damerow 1996, Divale 1999). The trajectories that emerge from this research, such as the unilinear sequences developed by Dehaene (1997), are unproven. My research on numerical notation showed that such trends complex and multilinear, correlate poorly with broad stages of social organization, and often relate instead to complex sociopolitical factors. By comparing the developmental trajectories of graphic and linguistic number representations, I will analyse the connection between them.

5) How do the cognitive and structural features of these two systems relate to the functions for which they are used? No study has ever examined the contexts in which written lexical numerals and numerical notation are preferred. If the differences in the properties of these two systems are as great as the evidence suggests, the choice of which system is used in specific contexts is extremely important. It is important to know in what contexts different forms of expression are used for the same number (e.g. 87 or LXXXVII versus eighty-seven or four score and seven). The formal properties of these representational systems probably influence the contexts in which they are used. I will examine in detail, for a large number of cases, the contexts in which numerical notation is used and those for which written lexical numerals are preferred.

It is very rare for a project that relies on anthropological, archaeological, historical, and linguistic data to contribute directly to ongoing work in cognitive studies and at the same time illuminate the relation between two aspects of culture in many different societies. Using a diachronic comparative framework, I can compile the data needed to answer theoretical questions relating to several disciplines and provide a database from which future projects can be launched.

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