Numerical Notation
Numerical Notation: A Comparative History (Cambridge, 2010)
Stephen Chrisomalis, Wayne State University
Numerical Notation is a cross-cultural anthropological examination of systems of written numerals used over the past 5000 years. I argue that numerical notations (like Roman numerals or Egyptian hieroglyphic numerals) are mainly social and representational tools, rarely if ever used directly for arithmetic. Understanding how they develop, change, and decline requires us to analyze how their users actually perceived and used them. Computational efficiency is only rarely relevant to their transformation and replacement. There are regularities and patterns among the world’s written numerals, but they derive from more general principles of representation and cognition. It is a thoroughly interdisciplinary study incorporating theories and principles from anthropology, archaeology, linguistics, epigraphy/paleography, cognitive science, developmental psychology, semiotics, and history of science.
Numerical Notation is now available for purchase through the Cambridge University Press website (click the image or link above), and is frequently available at a 20% or greater discount from Amazon.com
Reviews
The following scholarly reviews of Numerical Notation have appeared. I will list all reviews I know about, positive or negative.
Davis, Philip. 2010. SIAM News 43(5).
Aveni, Anthony. 2010. Journal of Anthropological Research 66 (4): 556-557.
Lloyd, G.E.R. 2010. Isis. 101 (4): 864-865.
Cuomo, S. 2011. Antiquity 85(328): 662-665.
Errata and Addenda
I sometimes tell my students that everything they ever do as scholars will be flawed, just as everything I’ve ever done or will ever do will be flawed. Meant to shock? Yeah, a little. But it’s worth remembering that this is the process of scholarship, error followed by correction. I’m sure that Numerical Notation has its share of boneheaded typos and brain-os, as well as corrections of fact or interpretation that can be handled in limited space. As readers contact me to note them, I’ll correct these below. Major re-evaluations will have to wait for an as-yet-extremely-hypothetical second edition.
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On p. 237 and again on p. 408 I allude to an explanation for the diversity of proto-cuneiform numerals that I thought I had made explicitly, but apparently not. One possibility is that there are multiple systems, not because any one scribe was jumping between many different systems for different commodities, but because there were partly distinct scribal traditions, each one requiring its users to use only one numeral system. If future archaeologists excavated Detroit and found a document with a bunch of hexadecimal numbers, another with times in hours:minutes:seconds, and a third with Western numerals, they might conclude that we had no unified numerical system. It gets a little more complicated in the case of proto-cuneiform (because texts with different systems have been found in the same locations, suggesting they were related to one another in antiquity) but it is, I think, an interesting hypothesis.
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Theodore Widom and Dirk Schlimm have proposed a revision to the typology I use in Numerical Notation. Its main improvement is that it disambiguates multiplicative-additive systems whose multipliers are expressed cumulatively from those expressed with ciphers (e.g. II M vs. 2K as two expressions for 2000). I consider this to be a very interesting approach. They presented their paper at the Society for Anthropological Sciences at my invitation in February 2011, and their article is forthcoming in Science in Context (I will post more when it is published).
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On page 127 I describe the Italian mathematician Luca Pacioli as an ‘accountant’. While it is certainly the case that that Pacioli’s Summa Arithmetica contains a great deal of information about accounting, including the first full description of double-entry bookkeeping, Pacioli was not a merchant or bookkeeper, but a mathematical scholar and Franciscan friar who was a collaborator of Leonardo da Vinci.
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