The primary function of numerical notation is to communicate numerical values. One cannot even lie effectively about how many enemies were killed in battle if the numerals being used are incomprehensible to the intended audience. Any attempt to explain the history of numerals without reference to the cognitive features underlying their structure is doomed to failure. Nevertheless, considerations of efficiency are not the sole or even the primary factor in the cultural evolution of numerical notation. While synchronic regularities may be explainable without reference to social context, diachronic regularities are not. Every cognitive advantage associated with a system is associated with disadvantages. The role of various social factors in explaining the history and development of numerical notation systems differs from case to case, depending on historical context, but they are always there. We cannot explain the replacement of Maya numerals by Western ones without consideration of the enormous social, political, and technological upheavals that were associated with the Spanish conquest of Mesoamerica. Numerical notation systems never exist as objects in isolation; their utility is not merely a function of their structure. By exploring the social contexts in which the transformation and replacement of numerical notation systems occur, it will be possible to evaluate the impact of social factors relative to purely cognitive and structural ones.

I have identified seventeen factors that influenced the changes in numerical notation systems examined throughout this study, any of which may apply to a particular historical event.