Category: Numerals

Last month, at the Society for Anthropological Sciences annual meeting in Charleston, SC, I organized a panel of some really interesting material on the broad topic of numeration. I want to take this opportunity (again) to thank all the presenters for their attendance and hard work. The abstracts (as also published in the conference program) were as follows:

Toward a cognitive, historical, linguistic anthropology of numerals
Stephen Chrisomalis (Wayne State University)

For over a century, the study of numeration, number systems and allied topics has been a key part of the comparative study of thought, language, and culture. The anthropology of numbers and mathematics has traditionally been a locus for unilinear evolutionary thought linked to notions of primitivity. The papers in this panel constitute a call for a culturally-grounded cognitive science of numeration within four of the disciplines of cognitive science (anthropology, linguistics, philosophy, and psychology).

Recent research in language evolution, linguistic relativity, and cultural aspects of mathematical cognition draw attention to the need for anthropologists to re-engage with this new agenda. First, the cross-cultural study of numerals allows the investigation and evaluation of universal and particular aspects of numeration and their relationship with social organization. Because numerals have multiple modalities (e.g., verbal, graphic, gestural), examining patterns in number systems beyond linguistics allows us to evaluate to what extent number concepts can be separated from language, including universal grammar. Finally, just as the cognitive anthropology of plant and animal taxonomy contributes to ecological and environmental anthropology, the cognitive anthropology of numerals and mathematics underpins economic anthropology and the anthropology of science.

Spatial-numeric associations in literates and illiterates
Samar Zebian (Lebanese American University, Beirut Lebanon)

Several independent studies have reported a cognitive association between small numbers and the left side of space and larger numbers andthe right side of space among individuals who read and write from left-to-right (SNARC effect). These associations are reversed for individuals who read and write from right-to-left. The SNARC effect has widely been taken as evidence that numbers are conceptualized as points along a mental number line, however there is growing evidence that this systematic spatial performance bias related to writing directionality is an instance of strategic processing rather than a reflection of inherent spatial attributes of numbers. In an attempt to explain the “deeper” origins of these associations researchers are examining the linkages between number and finger counting. The current study examines whether finger counting practices reveal consistent spatial-numeric associations and whether there are any spillover effects to other tasks that involve object sorting and counting and other non-counting but quantitative tasks such as line bisection and speeded parity judgment. If, in fact, finger counting practices and not the directionality of writing set up spatial-numeric associations than we should be able to observe the same type of spatial biases in literates and illiterates. Preliminary evidence suggests that the finger counting practices of literates and illiterates are not same and furthermore that the spatial biases found in finger counting are not observed across tasks.

Zero’s beginnings: the Mayan case
John Justeson (SUNY, Albany)

This paper addresses linguistic and (Mayan) historical evidence concerning the origins of a numerical concept of zero. Comparative linguistic evidence suggests that zero is not part of basic numerical cognition; rather, it develops out of computing practices of mathematical specialists. Specifically, while zero is often assumed to be prerequisite to the invention of positional notation, it seems on the contrary to emerge as a notational device within such systems. This is clearly the case in Mesoamerica. A system of place-value notation arose in Guatemala and Mexico among Mayans and epi-Olmecs by 36 BCE, with no symbol corresponding to a zero coefficient. Although data is limited, circumstantial evidence is consistent with the following scenario for the emergence of a numerical zero: Mayan calendar specialists developed discourse practices, associated with calendrically-timed ritual events, that used the word “lacking”; the associated dates were represented in a new, non-positional system of notation, which replaced positional notation except in calculating tables; the sign for “lacking” was transferred from the new notation into these tabular positional notations; as a side effect of the algorithms that specialists used to add and subtract positional numerals, the “lacking” symbol was reinterpreted numerically.

Methodological reflections on typologies for numeral systems
Theodore R. Widom and Dirk Schlimm (McGill University)

Past and present societies worldwide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. In the present paper we provide a general framework
within which the efficacy of these typologies can be assessed relative to certain desiderata. Using this framework, we discuss the two influential typologies of Zhang & Norman and Chrisomalis, and present a new typology which takes as its starting point the principles by which numeral systems represent multipliers (the principles of cumulation and cipherization), and
bases (those of integration, parsing, and positionality). We argue with many different examples that this provides a more refined classification of numeral systems than the ones put forward previously. We also note that the framework can be used to assess typologies not only of numeral systems, but of many domains.

Social relationships as a lexical source for numeral terms in Amazonia
Cynthia Hansen and Patience Epps (The University of Texas at Austin)

Due to the relatively high degree of etymological transparency found in the numeral systems of Amazonia, it is possible to see the range of lexical sources from which the numeral terminology emerges. In this paper, we present the range of strategies used to create numeral terms below 5, based on an extensive survey of the numeral systems of close to 200 Amazonian languages conducted by the authors. More specifically, we discuss a strategy that is well-attested in Amazonia but that is not attested elsewhere in the world: a ‘relational’ strategy where terms for 4 (and sometimes 3-10) are built using a social relationship term, such as ‘sibling’ or ‘companion’. We propose that this strategy mirrors a gestural counting strategy found throughout the region where fingers are grouped in pairs.

Cultural variation in numeration systems and their mapping onto the mental number line
Andrea Bender and Sieghard Beller (University of Freiburg)

The ability to exactly assess large numbers hinges on cultural tools such as counting sequences and thus offers a great opportunity to study how culture interacts with cognition. To obtain a more comprehensive picture of the cultural variance in number representation, we argue for the inclusion of cross-linguistic analyses. In this talk, we will briefly depict the specific counting systems of Polynesian and Micronesian languages that were once derived from an abstract and regular system by extension in three dimensions. The linguistic origins, cognitive properties, and cultural context of these specific counting systems are analyzed, and their implications for the nature of a (putative) mental number line are discussed.

Recently, there has been a “Puzzle Moment” in the science section of the New York Times, with an eclectic mix of articles combining scientific pursuits with cognitive and linguistic play of various sorts. One that caught my eye is ‘Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus’ by Pam Belluck [1], which is an account of the well-known Rhind Mathematical Papyrus. The RMP is an Egyptian mathematical text dating to around 1650 BCE, and is one of the most complete and systematic known accounts of ancient Egyptian mathematics. It’s a fascinating text, written in the Egyptian hieratic script rather than the more famous hieroglyphs, and it gives us considerable insight into the economy, social organization, and technical practices of the Second Intermediate Period.

The central conceit of the Times article is that the well-known “As I was going to St. Ives” poem-puzzle has its earliest ancestor in the RMP. This is vaguely true in that the RMP has a section involving repeated multiplication by seven, resulting in an addition problem. But Ahmes the scribe, despite his insistence that his text would reveal “obscurities and all secrets”, was not writing a mystery, but an exercise that formed part of scribal training, in an era where the literacy rate was at most 1-2%. While one can argue fairly that this is not a ‘real’ problem, and that the structure of it is meant to hold the learner’s attention through its repetitions, to call it a puzzle is only true in the broadest possible sense.

I’m a professional numbers guy, not an Egyptologist, but the article we are presented with not only tells us nothing new about the Rhind. I was very pleased, on the one hand, to see Marcel Danesi, whose work may be familiar to many readers of this blog, commenting on the widespread cross-cultural and cross-historical interest in puzzles (not only numerical puzzles, but including them). It’s not often enough that linguistic anthropologists get quoted in the Times. And like Danesi, I have broadly universalist sympathies. But I disagree with Danesi, who has made this claim about the RMP elsewhere, in his The Puzzle Instinct (Indiana, 2004, pp. 6-7) that it was “shrouded in mystery” or that “mystery, wisdom, and puzzle-solving were intrinsically entwined in the ancient world.”

The better example of numerical play in Egyptian scribal traditions mentioned in the Times article is the Horus eye, or wedjat, a combination of six symbols whose constituent parts signify the fractional series {1/2, 1/4, 1/8, 1/16, 1/32, 1/64} which when summed totals 63/64, or nearly one (see below). As the Egyptologist Sir Alan Gardiner reckoned it, the remaining 1/64 would be provided by Thoth who would heal the Eye and thus produce unity. It’s a nice story, and at least at some periods or for some writers, this narrative may have been relevant.

Source: wikimedia.org

But the Horus-eye illustrates one of the central problems in the transliteration of Egyptian texts, namely that while the vast majority of Egyptian mathematically-relevant texts are written in the cursive hieratic script, they are transcribed, and all-too-frequently theorized, as if they were hieroglyphs. This transcriptional practice leads us to think of the Rhind as a hieroglyphic text that just happens to be in hieratic in the original, but in the case of the Horus eye it couldn’t be more misleading. The Horus symbols in the Rhind don’t look like the above image, and more generally, the hieratic numerals look nothing like, and behave nothing like, the hieroglyphic numerals. We now call of these six Horus eye components by the less evocative name of ‘capacity system submultiples’ in recognition of the fact that these components were originally nonpictographic, part of a metrological system of grain measurement, and only at a much later date were they composed into the wedjat-eye. This isn’t to say that the Egyptians weren’t numerically playful, but they weren’t especially playful in the Rhind.

In short, the RMP is not an especially good example of numerical play in Egypt, and certainly not an especially relevant example from a cross-cultural perspective. It illustrates, to be sure, that mathematical texts are not purely functional or economic documents, but include semiotic and linguistic elements far beyond their pragmatic use. But this is not new knowledge about the Rhind or about mathematics. And it runs a grave risk of othering a document whose function was largely pedagogical, and is thus not so different than, for instance, the ‘ready-reckoners’ of early-capitalist sixteenth-century England.

I am thrilled to see numerical texts treated as objects of inquiry beyond the facile ‘Did they get the answer right?’ I am sympathetic to Danesi’s claim that puzzles and riddles have universal salience. Yet I worry that, at least in the case of the Rhind, the link to puzzle-like behavior is so far-fetched that it turns our best glimpse into Egyptian sociomathematical practice into an inappropriately arcane and obscurantist account. This ‘mysteries of lost Egypt’ nonsense should have been set aside decades ago.

If you wanted to pull out some cross-cultural examples of numerical play, you could easily find lots of better examples, from well-covered territory such as Hebrew gematria practices, to the richly evocative varnasankhya systems of number-word associations in premodern South Asian texts, to the complex cluster of quasi-cryptographic numerical systems used by Ottoman administrators and military officers. Or if you were really stuck on Egypt, you could investigate the cryptic numerals used on late Egyptian votive rods and Ptolemaic inscriptions, richly infused with homophony. (For a more extensive discussion of these and others, see my Numerical Notation: A Comparative History (Cambridge, 2010). There is a rich, although disciplinarily diverse, comparative body of material on numerical practices including puzzles, but the Rhind just isn’t part of it.

(Crossposted to the Society for Linguistic Anthropology blog)