Abstract
Drawing on the material culture of the Ancient Near East as interpreted through Material Engagement Theory, the journey of how material number becomes a conceptual number is traced to address questions of how a particular material form might generate a concept and how concepts might ultimately encompass multiple material forms so that they include but are irreducible to all of them together. Material forms incorporated into the cognitive system affect the content and structure of concepts through their agency and affordances, the capabilities and constraints they provide as the material component of the extended, enactive mind. Material forms give concepts the tangibility that enables them to be literally grasped and manipulated. As they are distributed over multiple material forms, concepts effectively become independent of any of them, yielding the abstract irreducibility that makes a concept like number what it is. Finally, social aspects of material use—collaboration, ordinariness, and time—have important effects on the generation and distribution of concepts.
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Notes
In actuality, this view is also relatively recent, as the treatment of neuroanatomist Franz Joseph Gall and his work in the early nineteenth century reveals: Between 1802 and 1817, the Church banned Gall’s writing, had him expelled from jobs, cities, and countries, and reportedly excommunicated him for suggesting the brain had something to do with mental functioning (Gall 1835; Moscati 1832; Tovino 2007).
While the brain is undeniably important to cognition, materiality is essential to psychological processes like perception, categorization, and abstraction. This means it is difficult to separate the internal and external aspects of cognition and leaves open the question of how much of the external world the brain reproduces internally (Hutto and Myin 2013).
Pictograms resembled what they meant (e.g., a picture of a head meant head because it looked like a head), while ideograms conveyed meaning through social agreement (e.g., a quartered circle meant sheep, not because it resembled a sheep in any way but because the Mesopotamians agreed that sheep is what the symbol meant).
Russell’s definition of number as cardinality shared by sets of objects implies that such concepts are realized by comparing sets for their cardinality, and indeed, such comparisons are well documented in the ethnographic literature as behaviors like pairing and one-to-one correspondence. In pairing, small sets of objects are compared; comparing two pairs affords an opportunity to appreciate their shared quantity (i.e., “two”), while comparing a pair with a single object affords an opportunity to appreciate their dissimilar quantity (i.e., “two” and “one”). In one-to-one correspondence, larger sets are iteratively matched, element by element, until all elements have been matched once with none remaining. Neither pairing nor one-to-one correspondence need presuppose concepts of or words for numbers because “it is simpler logically to find out whether two collections have the same number of terms than it is to define what that number is” (Russell 1920, p. 15). Certainly, “although language transforms and extends our conceptual abilities, it does not make them possible in the first place” (Parthemore 2013, p. 169). However, through mechanisms like enactive signification and pattern recognition, behaviors that manipulate objects into arrangements where their shared (or dissimilar) quantity can be recognized have the potential to generate such concepts.
Patterns are so important in numbers that mathematics has even been called the science of patterns (Devlin 2003).
There was a possible fourth indigenous counting system, ternal counting, that has been variously explained as a women’s language (an emesal or dialect spoken chiefly by women) or a literary convention (Whittaker 2002). However, ternal counting is associated with topics, archiving, and training found in scribal and priestly contexts (i.e., predominantly or exclusively male) (Lambert 1969), calling into question its identification as a woman’s dialect; it also seems overly specific to have been invented for purely literary purposes. A plausible alternative explanation is that it might have been the counting system of a minority culture.
Linguistic evidence shows that Mesopotamian peoples had the same perceptual experience of quantity that extant peoples do (Overmann 2015, 2016b). This is entirely predictable, given that the ability to appreciate quantity is found in nonhuman primates, mammals generally, birds, amphibians, fish, and perhaps even insects, phylogenetic distribution that suggests that the ability would undoubtedly have been shared by the species and peoples ancestral to extant humans. However, it is important to distinguish the perceptual experience of quantity from the concept of number, which is culturally constructed (Núñez 2017) through material engagement (Overmann 2016c).
The numerical impressions of the mid-fourth millennium BCE were organized by increasing magnitude, implying that the Neolithic tokens to which their sizes and shapes corresponded most likely were as well. This organization further implies that the surfaces on which tokens were manipulated may have been organized in some fashion, perhaps temporarily and as needed by drawing lines in dust or sand, or more formally and repeatably through the use of counting boards. The word abacus may have originated in the West Semitic word for dust (Semitic Roots2017), and Akkadian was a related (East Semitic) language. Further, the Sumerian signs for words like counting had forms that were vaguely abacus-like, and abaci (and before them, counting boards with pebbles or calculi) were known in many parts of the ancient world (Ifrah 2000). These circumstances have led authors like Ifrah (1981, 2000) to infer that the abacus was used in Mesopotamia. Certainly, the abacus provides both visual discriminability and manipulability. However, the use of an abacus in Mesopotamia has not been established through archaeological or textual means (e.g., findings of abaci or their depictions, like the Salamis counting board and Dareios vase from the first millennium BCE; mentions of abaci, counting boards, lines in the dust, etc., or descriptions of methods used to organize the surfaces used for manipulating tokens, like those recounted by Polybius in the second century BCE).
Whether numbers are conceived as collections or entities is linguistically distinguished: In collections, two and two are four, while with entities, two plus two is four (Gowers 2008).
Numbers are translinguistic in their apprehension and communicability (this is a function of how materiality instantiates quantity, and possibly because the appreciation of quantity is an innate perceptual sense that humans share with other species), so numeracy in that sense is less limitable to an individual. For literacy, the distinction is being drawn with regards to a modern script, which cannot be read without the requisite training and practice; writing systems that involve pictograms and ideograms (e.g., early Mesopotamian writing) require less training and practice, so in that sense they would be less limitable to an individual as well.
Accessibility means average people can acquire and participate in basic numeracy and literacy, not necessarily that everyone can perform advanced mathematics or engage the range of thought opened up by literacy.
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Acknowledgements
I thank two anonymous reviewers for their close reading of and insightful comments on the draft, which led to its improvement. Admittedly, I resisted their recommendations for greater inclusion of the social aspects of numerical cognition in this work. As an archaeologist, where many researchers in numerical cognition foreground social transactions and mention the material as an ancillary matter, I reverse this order specifically and intentionally to focus on how the material structures used for counting inform the content, structure, and organization of numerical concepts over time, as well as the use of materiality as a collaborative medium for change on the level of society. The extent to which this is a fault is entirely mine. I also thank Lambros Malafouris, whose work significantly influences my own, for the opportunity to submit a paper for this special issue.
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Overmann, K.A. Concepts and how they get that way. Phenom Cogn Sci 18, 153–168 (2019). https://doi.org/10.1007/s11097-017-9545-8
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DOI: https://doi.org/10.1007/s11097-017-9545-8