Stephen Chrisomalis, The Comparative History of Numerical Notation, Ph.D. dissertation (McGill University, 2003)
	Numerical notation systems are structured, visual, and primarily non-phonetic systems for representing number. This study employs a diachronic and comparative framework to examine over 100 systems used during the past 5000 years. The historical context of each system's origin, transmission, transformation, and decline is traced, linking systems together into phylogenies, but according priority to neither analogical or homological explanations. Structural aspects of numerical notation systems are compared and the limits of variability among them are established. A two-dimensional typology is presented that analyzes the intraexponential and interexponential structuring of each system, in addition to one or more numerical bases. In previous approaches, the only relevant factor considered was the presence or absence of positionality, which led inevitably to unilinear and progressivist conclusions. The analysis of historical relations among numerical notation systems permits a direct approach to questions of how and why they changed. The application of a multilinear cultural evolutionary framework reveals both synchronic and diachronic regularities among numerical notation systems. Where possible, these cross-cultural regularities are related to principles of cognitive psychology. Full explanations of the cultural evolution of numerical notation must also take account of social factors because changes in systems are always the product of decisions made in particular social contexts. Most numerical notation systems are used only for recording and communication, not computation, so it is illegitimate to evaluate their usefulness for functions for which they were never used. A model is presented that relates structural features of numerical notation systems to the contexts of their use and transmission. Because positional systems are most useful for functions related to administrative and scientific institutions that promote cultural hegemony, the observed trend towards positional numerals is a consequence of the dominance of societies that possess such institutions rather than the numerals' inherent superiority.
Stephen Chrisomalis, Ph.D. Copyright 2004