Mangarevan invention of binary steps for easier calculation

Andrea Bender; Sieghard Beller

doi:10.1073/pnas.1309160110

Edited by Russell D. Gray, University of Auckland, Auckland, New Zealand, and accepted by the Editorial Board November 15, 2013 (received for review May 14, 2013)

Significance

The paper describes the mixed counting system in Mangarevan, which is unique in that it had three binary steps superposed onto a decimal structure. In showing how these steps affect calculation, our analysis yields important insights for theorizing on numerical cognition: counting systems serve as complex cultural tools for numerical cognition, apparently unwieldy systems may in fact be cognitively advantageous, and such advantageous systems can be—and have been—developed by nonindustrialized societies and in the absence of notational systems. These insights also help to dismiss simple notions of cultural complexity as a homogenous state and emphasize that investigating cultural diversity is not merely an optional extra, but a must.

Abstract

When Leibniz demonstrated the advantages of the binary system for computations as early as 1703, he laid the foundation for computing machines. However, is a binary system also suitable for human cognition? One of two number systems traditionally used on Mangareva, a small island in French Polynesia, had three binary steps superposed onto a decimal structure. Here, we show how this system functions, how it facilitated arithmetic, and why it is unique. The Mangarevan invention of binary steps, centuries before their formal description by Leibniz, attests to the advancements possible in numeracy even in the absence of notation and thereby highlights the role of culture for the evolution of and diversity in numerical cognition.

Footnotes

↵¹To whom correspondence should be addressed. E-mail: Andrea.Bender{at}psysp.uib.no.

Author contributions: A.B. and S.B. designed research, performed research, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission. R.D.G. is a guest editor invited by the Editorial Board.
*Descriptions of these systems vary, and our previous publications containing reference to the Mangarevan system (34, 35) were based on the description by the French padre Vincent-Ferrier Janeau (20). Meanwhile, however, we have reasons to believe that the description provided by Sir Peter Buck, also known as Te Rangi Hiroa (10)—an anthropologist, doctor, and politician of Polynesian descent—is the more meticulous and authoritative account (36).
^†It can be observed in Māori architecture and decoration, for instance, in the custom of putting even numbers of rafters on either side of a roof so as to avoid bad luck (42, 43), and Māori fowlers were said to have avoided obtaining odd numbers by simply waiting for more prey (44). On Hawai’i, four was of extreme significance in a spiritual context, and both four and eight were formulistic numbers (26, 41). The concern with symmetry is also reflected in the specific counting systems across Polynesia, almost all of which used even numbers such as 2, 4, 10, or 20 as counting units (33, 34).
^‡Besides its binary steps, the mixed system had another advantage, namely that the counting units are larger than 1 (mostly 2, 4, and 8). This increase in size mainly served to extract that very factor from the quantities to be counted or calculated (30, 33–35).