In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities (...) between the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793. (shrink)

The idea the New Zealand Māori once counted by elevens has been viewed as a cultural misunderstanding originating with a mid-nineteenth-century dictionary of their language. Yet this “remarkable singularity” had an earlier, Continental origin, the details of which have been lost over a century of transmission in the literature. The affair is traced to a pair of scientific explorers, René-Primevère Lesson and Jules Poret de Blosseville, as reconstructed through their publications on the 1822–1825 circumnavigational voyage of the Coquille, a French (...) corvette. Possible explanations for the affair are briefly examined, including whether it might have been a prank by the Polynesians or a misunderstanding or hoax on the part of the Europeans. Reasons why the idea of counting by elevens remains topical are discussed. First, its very oddity has obscured the counting method actually used—setting aside every tenth item as a tally. This “ephemeral abacus” is examined for its physical and mental efficiencies and its potential to explain aspects of numerical structure and vocabulary (e.g., Mangarevan binary counting; the Hawaiian number word for twenty, iwakalua), matters suggesting material forms have a critical if underappreciated role in realising concepts like exponential value. Second, it provides insight into why it can be difficult to appreciate highly elaborated but unwritten numbers like those found throughout Polynesia. Finally, the affair illuminates the difficulty of categorising number systems that use multiple units as the basis of enumeration, like Polynesian pair-counting; potential solutions are offered. (shrink)

We present an algebraic account of the Tongan kinship terminology (TKT) that provides an insightful journey into the fabric of Tongan culture. We begin with the ethnographic account of a social event. The account provides us with the activities of that day and the centrality of kin relations in the event, but it does not inform us of the conceptual system that the participants bring with them. Rather, it is a slice in time of an ongoing dynamic process that links (...) behavior with a conceptual system of kin relations and vice versa. To understand this interplay, we need an account of the underlying conceptual system that is being activated during the event. Thus, we introduce a formal, algebraically based account of TKT. This account brings to the fore the underlying logic of TKT and allows us to distinguish between features of the kinship system that arise from the logic of TKT as a generative structure and features that must have arisen through cultural intervention. (shrink)