Counting and the ontogenetic origins of exact equality

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.

The Material Origin of Numbers (front matter and introduction)

The Material Origin of Numbers examines how number concepts are realized, represented, manipulated, and elaborated. Utilizing the cognitive archaeological framework of Material Engagement Theory and culling data from disciplines including neuroscience, ethnography, linguistics, and archaeology, Overmann offers a methodologically rich study of numbers and number concepts in the ancient Near East from the late Upper Paleolithic Period through the Bronze Age. This project has received funding from the Clarendon Fund at the University of Oxford, as well as the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

Number Concepts Are Constructed through Material Engagement: A Reply to Sutliff, Read, and Everett

I respond to three responses to my 2015 Current Anthropology article, “Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number.” This study examined the categorical and geographical distribution of lexical numbers, also known as counting numbers, and grammatical number, the ability to linguistically distinguish singular and plural. Both these features of language conform to the perceptual experience of quantity, which consists of subitization, the ability to rapidly and unambiguously identify one, two, and three, and magnitude appreciation, the ability to appreciate bigger and smaller in the numerical quantity of groups when the difference lies above a threshold of noticeability. My reply to Sutliff disagrees with her contention that mathematical ideas are mentally innate on the grounds that this ignores their explicit construction through the interaction of human psychological, physiological, and behavioral abilities with materiality. My reply to Read expands on the idea that language obscures cross-cultural conceptual variability in number concepts because everything that translates as “three” does not necessarily have the same numerical properties. Finally, my reply to Everett notes that investigating numerical origins means discarding the deeply entrenched assumption of linguistic primacy on the grounds that material forms make numerical intuitions tangible, visible, and manipulable in ways that language cannot and, moreover, provide an alinguistic bootstrap mechanism that accounts for the emergence of both concepts of number and words for the concepts.

Material Scaffolds in Numbers and Time

The present paper develops a framework for interpreting Upper Paleolithic artifacts from an analysis of material complexity, numeration systems, and timekeeping using cultural categorizations (Hayden & Villeneuve, 2011), insights on the emergence of number terms in language (Menninger, 1992), and the astronomy practices of 33 contemporary hunter–gatherer societies (Yale’s Electronic Human Relations Area Files World Cultures database). Key findings: (1) an absence of societies with minimal material complexity and later-stage numeration systems, suggesting that material scaffolding may be important to realizing explicit number concepts, (2) the consistency of material complexity with both early- and later-stage numeration systems, emphasizing that material complexity may precede and inform the development of complexity in numeration systems, (3) the compatibility of astronomical practices with the spectrum of complexity in material culture and numeration systems, suggesting that the awareness of time may precede both, and (4) the increasing quantification of time consistent with greater material and numeration complexity, suggesting the availability of numbers as a cognitive technology may enable the structuring of time. These findings suggest that astronomy originates in the ability to estimate and infer contextual relations among natural phenomena and transitions from these natural associations to material representations and cognitive technologies that mediate conceptual apprehensions of time as a substance that can be quantified. Given that artifacts may act as scaffolds for explicit concepts of numbers and numbers scaffold explicit concepts of time, prehistoric artifacts such as the Blombos Cave beads (ca. 75,000), Abri Blanchard and Cellier artifacts (ca. 28,000), and Taï plaque (ca. 14,000) may represent similar scaffolding and conceptual development in numbers and time. It is proposed that the prehistoric societies making these artifacts possessed, in addition to material complexity, the abilities to express quantities in language and to use material externalization and cognitive technologies. Further, the Abri Blanchard artifact is proposed to represent externalized working memory, a very modern interaction between mind and material culture.