The Material Origin of Numbers: Insights from the Archaeology of the Ancient Near East 9781463240691

What are numbers, and where do they come from? These questions have perplexed us for centuries, if not millennia. A nove

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The Material Origin of Numbers: Insights from the Archaeology of the Ancient Near East
 9781463240691

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Gorgias Studies in the Ancient Near East

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6HULHV(GLWRULDO%RDUG 5RQDOG:DOOHQIHOV 3DXO&ROOLQV $LGDQ'RGVRQ $OKHQD*DGRWWL .D\.RKOPH\HU¬ $GDP0LJOLR %HDWH3RQJUDW]/HLVWHQ 6HWK5LFKDUGVRQ

This series publishes scholarly research focusing on the societies, material cultures, technologies, religions, and languages thatemerged from Mesopotamia, Egypt, and the Levant. *RUJLDV6WXGLHVLQWKH$QFLHQW1HDU(DVW features studies with bothhumanistic and social scientific approaches. 

The material origin of numbers

Insights from the archaeology of the Ancient Near East

Karenleigh A. Overmann

gp 2019

Gorgias Press LLC, 954 River Road, Piscataway, NJ, 08854, USA www.gorgiaspress.com Copyright © 2019 by Gorgias Press LLC All rights reserved under International and Pan-American Copyright Conventions. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise without the prior written permission of Gorgias Press LLC.

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2019

ISBN 978-1-4632-0743-4

Cover. Mathematical tablet (HS 201) from Nippur. It dates somewhere between the Ur III (2100–2000 BCE) and Early Old Babylonian (2000–1900 BCE) periods and is part of the Frau Professor Hilprecht Collection, University of Jena, Germany. The table is a list of reciprocals, an excellent example of the relational data transforming the concept of number. In 1935, Otto Neugebauer read the last line of the right column as 32 being the reciprocal of 112.5 (‘32 igi 1,52,30’, Mathematische Keilschrift-Texte, p. 10). The photo was produced by Manfred Krebernik and his assistant, and use of the image is courtesy of the Frau Professor Hilprecht Collection, University of Jena, Germany.

A Cataloging-in-Publication Record is available from the Library of Congress. Printed in the United States of America

The material origin of numbers: Insights from the archaeology of the Ancient Near East

Do not delete the following information about this document. Version 1.0 Document Template: Template book.dot. Document Word Count: 12772 Document Page Count: 323 To Bill: My first, my last, my always, and everything in between.

TABLE OF CONTENTS Table of Contents ................................................................................................................ v Figures and Tables ............................................................................................................. vii Figures ........................................................................................................................ vii Tables ......................................................................................................................... viii Conventions ......................................................................................................................... ix Timeline ....................................................................................................................... ix Abbreviations for museum designators.................................................................. ix Acknowledgements ............................................................................................................. xi Chapter 1. Introduction ...................................................................................................... 1 Chapter 2. Numbers through a different lens ................................................................. 9 Cognition is extended and enactive ......................................................................... 9 Materiality has agency ............................................................................................... 14 Material signs are enactive ....................................................................................... 19 Numbers through a new lens .................................................................................. 24 Chapter 3. What’s a number, really? ............................................................................... 25 An archaeology of number and a role for material forms .................................. 30 Two useful theoretical constructs .......................................................................... 32 Chapter 4. Assembling an elephant, one bit at a time .................................................. 43 Numerosity, the sense of quantity .......................................................................... 43 Language and numbers ............................................................................................ 46 Development and numbers ..................................................................................... 49 Categorization and abstraction ............................................................................... 53 Finger-counting and human neuroanatomy ......................................................... 56 Chapter 5. Behavioral traces............................................................................................. 65 A caveat about the historic ethnographic literature............................................. 67 Peoples count the same way.................................................................................... 70 Questions we might ask at this point..................................................................... 85 Chapter 6. Language in holistic context ......................................................................... 89 Lexical numbers ........................................................................................................ 93 Grammatical number ............................................................................................. 100 Ordinal numbers ..................................................................................................... 102 What to look for in the Mesopotamian languages, and why............................ 104

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Chapter 7. Ancient languages and Mesopotamian numbers ..................................... 107 The ancient languages of Mesopotamia: Sumerian, Akkadian, and Elamite ..................................................................................................... 110 Evidence of numerical language from ancient writing ..................................... 112 Lexical numbers ...................................................................................................... 115 Grammatical number ............................................................................................. 121 Ordinal numbers ..................................................................................................... 123 Wrapping up the evidence of the ancient languages ......................................... 127 Chapter 8. Fingers and tallies ......................................................................................... 131 The why, how, and what of finger-counting ...................................................... 133 Everything you wanted to know about tallies .................................................... 140 Evidence of tallies in the Ancient Near East...................................................... 146 The context for evaluating artifacts for use as tallies ........................................ 152 Chapter 9. The Neolithic clay tokens ........................................................................... 157 Tokens and numerical meaning ............................................................................ 161 Issues in interpreting tokens as numerical counters .......................................... 164 Newly catalogued token finds and their analysis ............................................... 166 The complexity of tokens and token-based accounting ................................... 174 Chapter 10. Numerical notations and writing ............................................................. 179 From tokens to impressions.................................................................................. 180 From impressions to commodity labels .............................................................. 184 How early Mesopotamian writing became the cuneiform script .................... 187 The effects of writing on numbers....................................................................... 196 More effects of writing on numbers .................................................................... 201 Chapter 11. The role of materiality in numerical concepts ....................................... 207 The sequence of material forms used for counting ........................................... 209 Other materially influenced change ..................................................................... 217 Distribution, independence, and other so-called abstract qualities................... 221 Chapter 12. Concluding remarks and questions ......................................................... 229 Some answers to the questions posed ................................................................. 232 Directions for future research ............................................................................... 241 Appendix: Data tables ..................................................................................................... 245 Bibliography ...................................................................................................................... 257 Index .................................................................................................................................. 297

FIGURES AND TABLES FIGURES 3.1 4.1 4.2 4.3 5.1 5.2 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 8.1 8.2 8.3 8.4 8.5 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 10.1 10.2 10.3 10.4 10.5 10.6

Conceptual blending with a material anchor Subitization and magnitude appreciation The cortical homunculus Topographical layout of the motor and somatosensory cortices One-dimensional devices Two-dimensional devices Frequency of use for the lexical numbers one through twenty in American English Distribution of 452 languages with analyzable number words Distribution of 168 languages with lexical numbers and grammatical number Ordinal frequency Sumer, Akkad, and Elam in the 3rd millennium BCE Lexical tablet with phonetic values for the Sumerian numbers two through ten Sumerian lexical numbers Sumerian ordinal frequency by writing state Sumerian ordinal frequency by document type Geographic distribution of the oldest sites with possible tallies Worked bones from Kebara and Ha-Yonim, Israel Worked bone from Ksar’Aqil, Lebanon Worked bones from Jita, Lebanon and Ain el-Buhira, Jordan Pace of technological invention, elaboration, and accumulation Correspondences between plain tokens, numerical impressions, and cuneiform number signs Clay bulla and early numerical impressions Chronology of artifacts used in Mesopotamian accounting and mathematics Types of tokens Temporal distribution of tokens Geographic distribution of early tokens Temporal distribution of plain types used as numerical counters The cuneiform sign ŠID Magnitude ordering of numerical signs Uruk V artifacts with higher-than-expected N14 repetition Geographic distribution of numerical tablets assigned to the Uruk V period Administrative tablet (W 6066,a) from the city of Uruk Representational modes of signs in early writing Chronology of signs vii

viii 10.7 10.8 10.9 10.10 10.11 11.1 11.2

THE MATERIAL ORIGIN OF NUMBERS Feature recognition of physical and written objects The development of literacy from writing Mathematical tablet (Erm 14645) Change in numerical and non-numerical signs Change in numerical signs Chronology of material artifacts used in Mesopotamian numbers The elaboration of numbers

TABLES 3.1 4.1 5.1 6.1 7.1 7.2 7.3 7.4 8.1 9.1 9.2 10.1 11.1 11.2 12.1 12.1 A.1 A.2 A.3

Russell’s logical types Categorical judgments Number words in four Yuki dialects Ordinal word frequency in American English, Mandarin Chinese, and Arabic Sumerian ternal counting and eme-sal numbers Akkadian lexical numbers Grammatical number in Sumerian, Akkadian, Elamite, and English Sumerian ordinal frequency by time period and document composition Early tallies Counting systems used with tokens Tokens by country and site Uruk V numerical tablets with exceeded bundling Affordances and limitations of material artifacts used for Mesopotamian numbers Chinese and English numbers Cuneiform numbers Comparison of numerical notations Pre-Uruk V (8500–3500 BCE) numerical impressions and tokens Uruk V (3500–3350 BCE) numerical impressions Newly catalogued tokens

CONVENTIONS TIMELINE Late Upper Paleolithic Epipaleolithic Neolithic Chalcolithic (Copper Age) Pre-Uruk V Uruk V Uruk IV Early Bronze Age Uruk III Jemdet Nasr (JN) Early Dynastic (ED) I/II Early Dynastic (ED) IIIa Early Dynastic (ED) IIIb Old Akkadian (OA) Lagaš II Ur III Early Old Babylonian Old Assyrian (Old Assyr.) Old Babylonian (OB) Middle Assyrian (MA) Neo-Assyrian (NA) Neo-Babylonian (NB)

30 to 12 thousand years ago 12,000 years before present to 8300 BCE 8300–4500 BCE 4500–3300 BCE 8500–3500 BCE 3500–3350 BCE 3350–3200 BCE 3300–2000 BCE 3200–3000 BCE 3200–3000 BCE 2900–2700 BCE 2600–2500 BCE 2500–2340 BCE 2340–2200 BCE 2200–2100 BCE 2100–2000 BCE 2000–1900 BCE 1950–1850 BCE 1900–1600 BCE 1400–1000 BCE 911–612 BCE 626–539 BCE

ABBREVIATIONS FOR MUSEUM DESIGNATORS A AO Ashm CUNES DV IM JA JRL MS, MT, MW

Oriental Institute, University of Chicago, Illinois, USA Louvre Museum, Paris, France Ashmolean Museum, Oxford, England Cornell University, Ithaca, New York, USA State Hermitage Museum, St. Petersburg, Russian Federation Iraq Museum, Baghdad, Iraq National Museum of Syria, Raqqa, Syria University of Manchester, England Private, anonymous collections in Europe ix

x NIM NMSDeZ NMSR OIM A Sb T UM VA, VAT W

THE MATERIAL ORIGIN OF NUMBERS National Museum, Tehran, Iran National Museum of Syria, Der-ez-Zor, Syria National Museum of Syria, Raqqa, Syria Oriental Institute, University of Chicago, Illinois, USA Louvre Museum, Paris, France National Museum of Syria, Damascus, Syria University of Pennsylvania Museum of Archaeology and Anthropology, Philadelphia, Pennsylvania, USA Vorderasiatisches Museum, Berlin, Germany Artifact found at the ancient city of Uruk in Iraq

ACKNOWLEDGEMENTS This undertaking would not have developed in the way it ultimately did without the massive investment of time, expertise, and encouragement by the scholars who have been my mentors, supervisors, professors, colleagues, influences, and friends, listed here in roughly chronological order: Joan Ray, Rex Welshon, Thomas Wynn, Frederick Coolidge, Chris Gosden, Andrea Bender, Steven Chrisomalis, Pierre Pica, Jerrold Cooper, and Robert Englund. Non-chronologically, as they require special thanks, are several individuals. Denise Schmandt-Besserat encouraged my interest in, and facilitating my access to, data on the Neolithic clay tokens, including her own extensive research. Jacob Dahl introduced me to Assyriology, taking me from the little I recalled from middle school—that Mesopotamia was the land between the Tigris and the Euphrates—to my present understanding. John MacGinnis and Tim Matney granted me complete access to their database of recent token finds from Ziyaret Tepe, generosity unparalleled among my experience with the many owners of such data. Two anonymous reviewers provided insight that helped me refine the thesis toward the present publication. Lambros Malafouris significantly influenced the direction of my thinking, while allowing me complete freedom to develop it as it made sense to me. Last and never least, Colin Renfrew gave me the single most useful piece of advice I received during my doctoral studies. I also thank the Clarendon Fund for the generous scholarship that supported my doctoral research at Oxford, and the European Union’s Horizon 2020 Programme for the Marie Skłodowska-Curie Actions individual fellowship grant funding my current postdoctoral research at the University of Bergen. Finally, I could not have persisted in this endeavor without the unstinting love and support of my family: Bill, my husband and best friend of nearly four decades; my son Archie, near to finishing his six-year residency in orthopedic surgery; my daughter Barbara, newly reported to the Pentagon for a fellowship in Air Force medical administration; and my granddaughters Jaiyah and Siena, now 15 and 1. It is with keen and unending sorrow that I note the loss of our other son Will, Archie’s twin, to a long and difficult battle with alcoholism in 2016. Having the escape of a thesis to finish, along with the kindness of the Keble College community in Oxford and our neighbors Robbie and Dana in Colorado, helped me endure an otherwise life-shattering tragedy. Karenleigh A. Overmann Department of Psychosocial Science University of Bergen, Norway May 2019 xi

CHAPTER 1. INTRODUCTION I had been at Oxford for only two weeks when my doctoral supervisor, cognitive archaeologist Lambros Malafouris, introduced me to an eminently distinguished and intimidatingly famous archaeologist, a scholar whose contributions to the field were so legendary they had been recognized not with mere knighthood but peerage. I speak, of course, of Colin Renfrew, Baron of Kaimsthorn, Fellow of the British Academy and the Society of Antiquaries, Emeritus Disney Professor of Archaeology at Cambridge University, Senior Fellow of the McDonald Institute for Archaeological Research, one of the leading pioneers of cognitive archaeology, and Lambros’ own doctoral supervisor. With a flattering interest, Renfrew asked me the topic of my thesis. Upon learning it was the clay tokens used as numerical counters in Neolithic Mesopotamia, he snapped, ‘Those have been done to death! Why study them? Do something else!’ His emphatic doubt about what, if anything, the least bit interesting might be left to say about these artifacts has lingered at the back of my mind the several years since, daring me to address it. Why study the Neolithic tokens? The question is highly pertinent, since indeed, the tokens have been investigated and published upon, extensively if not exhaustively, since archaeologists like Vivian Broman,1 A. Leo Oppenheim,2 and Pierre Amiet3 first noticed their shapes, sizes, and quantities corresponded to those of the clay impressions that preceded handwritten forms for numbers in the mid-to-late 4th millennium BCE. The tokens’ role in the invention of writing has long been proclaimed, they themselves painstakingly catalogued and analyzed, at least as much as they were known up until the early 1990s, by archaeologist Denise Schmandt-Besserat.4 Lambros himself5 had already explored and written about their significance as components of the cognitive system for numbers, the very inquiry I was proposing to take up yet again. What could possibly be added to such work that was worthy of the undertaking? And, if such Broman, Jarmo Figurines. Oppenheim, ‘On an Operational Device in Mesopotamian Bureaucracy’. 3 Amiet, Mémoires de la Délégation Archéologique en Iran, Tome XLIII, Mission de Susiane. 4 Schmandt-Besserat, Before Writing: From Counting to Cuneiform. 5 Malafouris, ‘Grasping the Concept of Number: How Did the Sapient Mind Move beyond Approximation?’. 1 2

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illustrious scholars had already discovered and said everything it was possible to know on the topic of tokens, what could a mere student bring to the inquiry? Here it’s worth noting I had a somewhat different perspective on numbers than that of these antecessors. For the several years prior to starting at Oxford, I had worked at the University of Colorado at Colorado Springs with psychologist Frederick Coolidge and archaeologist Thomas Wynn. They had introduced me to the idea that change in the form of material artifacts over time could indicate cognitive change in the species making the artifacts. Wynn, another leading pioneer of cognitive archaeology, had discerned change in the spatial cognition of species ancestral to humans through their increasing imposition of shape and symmetry on stone tools.6 Coolidge had brought to their partnership the idea of investigating the expansion of working memory through artifacts like traps that implied such executive functions.7 In my own work with Coolidge and Wynn, much of my research had focused on understanding how number systems emerge from the perceptual experience of quantity humans share with other species, and how this differs from language.8 I had considered the questions of what numbers are and how we get them through the methods and data of psychology, language, and ethnography: what the brain does in numbers, how language puts number words together, and how peoples in different cultures and societies count. I had also investigated the links between language for numbers, cultural complexity, and numerical complexity,9 though by the time I arrived at Oxford, I was becoming increasingly dissatisfied with these findings, since they had only concluded, as other disciplines had,10 that there was some kind of a link, the details of which could not be determined. Further, I had not yet given much thought to what material forms for representing and manipulating numbers had to do with numerical origins and elaboration,11 something Lambros encouraged me to do. Once I started reading the Assyriological literature, I had enough background in anthropological theory to be disquieted by the claim that numbers were somehow concrete before they became abstract. This assumption pervaded the work of both Schmandt-Besserat12 and Malafouris,13 as well as that of other scholars writing about numerical origins and developmental acquisition—psychologists Peter Damerow and

Wynn, The Evolution of Spatial Competence. Coolidge and Wynn, ‘Executive Functions of the Frontal Lobes and the Evolutionary Ascendancy of Homo sapiens’. 8 Coolidge and Overmann, ‘Numerosity, Abstraction, and the Emergence of Symbolic Thinking’. 9 Overmann, ‘Material Scaffolds in Numbers and Time’; Numbers and Time: A Cross-Cultural Investigation of the Origin and Use of Numbers as a Cognitive Technology; ‘Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number’. 10 Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’. 11 Overmann, ‘The Role of Materiality in Numerical Cognition’. 12 Schmandt-Besserat, ‘An Archaic Recording System and the Origin of Writing’. 13 Malafouris, ‘Grasping the Concept of Number’. 6 7

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Jean Piaget.14 Since the 1970s, cultural anthropology had more than frowned upon categorizing societies as either primitive or advanced, and as evolving from one to the other. Such proscriptions reacted to the discipline’s own origins in the work of 19thcentury scholars like Edward Burnett Tylor, who announced his opinion of nonWestern societies in the title of his most famous book: Primitive Culture.15 I had also encountered the idea of philosopher of mathematics Bertrand Russell that numbers are the recognition of cardinality shared between sets of objects.16 This seemed to me to be capable of bridging the gap between the perceptual experience of quantity and the conceptualization of number. These insights suggested the development of numerical concepts from the perceptual experience of quantity not only could but had to be explained without appealing to constructs that invoked inherently flawed and fortunately outdated assumptions about societal modes of thinking. Other matters that immediately confronted my understanding of what numbers were like were the Assyriological claims that tokens were not used with number words, and that they were the first material form used for counting. From the vantage of my admittedly meager knowledge and experience, I intuited these simply had to be incorrect. Firstly, when numbers initially emerge in language, they are limited to, and quite consistent with, the perceptual experience of quantity: one and two, possibly three, and occasionally four, with numbers above that range being many, a characteristic noted for number words around the world, across significant spans of time, and by observers from vastly different fields.17 No known number system uses material forms to represent numbers in the hundreds and thousands, as the Mesopotamian tokens did at the point where impressions provide insight into how the tokens were used, while concomitantly lacking a comparable lexicon of number words; similarly, no known numerical lexicon lags behind the values represented materially to the extent being claimed for Mesopotamian numbers. These things suggested there would have been a numerical lexicon, even if the evidence for it was both limited and much later. Secondly, although tokens may look rudimentary when viewed from the perspective of the complexity that mathematics reach by the Old Babylonian period (1900– 1600 BCE), they do not from the perspective of how numbers first emerge. I had been studying the vocabularies, behaviors, and devices of traditional peoples taking their first, early steps into numbers; the tokens in comparison were overwhelmingly complex, as was the material culture of urbanized Mesopotamian agriculturalists when contrasted with that of nomadic hunter–gatherers. For the tokens, this complexity lay 14 Damerow, ‘Number as a Second-Order Concept’; ‘Prehistory and Cognitive Development’; Piaget, The Child’s Conception of Number; ‘Logique Génétique et Sociologie’. 15 Tylor, Primitive Culture: Researches into the Development of Mythology, Philosophy, Religion, Art, and Custom. 16 Russell, Introduction to Mathematical Philosophy; ‘The Theory of Logical Types’. 17 Conant, The Number Concept: Its Origin and Development; Greenberg, ‘Generalizations about Numeral Systems’; Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer; Menninger, Number Words and Number Symbols: A Cultural History of Numbers; Ore, Number Theory and Its History; Tylor, Primitive Culture.

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in their being related to each other, with some quantity of lower-value tokens being represented by and exchanged with a single token of higher value. This characteristic was something other than one-to-one correspondence, and it implied that tokens would have been preceded by other, simpler, technologies, with forms and structures that could explain how the first numbers consistent with the perceptual experience of quantity might eventually yield the relatively complex numerical concepts represented by tokens and later notations. Part of any investigation, then, needed to address the question of precursor technologies, in terms of what might constitute evidence for them and how they might influence numerical content, structure, and organization. Ultimately, my attempts to reconcile the Assyriological assumptions with my prior anthropological, psychological, and linguistic understandings of numbers provided me an opportunity to say something new, not just about the Neolithic tokens in Mesopotamian numbers but also about the role of material structures in human cognition more generally. This work, an updated version of my doctoral thesis, presents my view of how the material structures used to represent and manipulate numbers inform their content, structure, and organization: whether numbers are conceptualized as equivalences, collections, or entities; whether and to what extent they possess properties like linearity and magnitude ordering; and how closely tied they are as concepts—or not— to particular material structures. Essentially, I view numbers as abstract from their inception and materially bound at their most elaborated. This view collapses the historical distinction between the so-called concrete and abstract modes of thinking. It also provides an elaborational mechanism, namely, the incorporation of additional material forms, that explains why and how number concepts change over time, reveals their inherent similarities and differences across cultures at a time and within societies over time, and explains how societies comprised of average individuals can be capable of realizing complex cultural systems like numeracy and mathematics in the first place. Mesopotamia proved an ideal case study. It has an unusually long and detailed sequence of material devices used for counting—if one admits evidence from a variety of sources and then infers from it freely. Admittedly, such inferences have the potential to irk those of a historical bent, but they are not uncommon in archaeology. Textual evidence, which provides insight into the five-plus formations of the Sumerian numbers six, seven, and nine, suggests finger-counting, an assessment based on the presence and similar interpretation of identically compounded number words in other languages. This evidence occurs more than a thousand years after the invention of writing, and there is no known way of determining exactly when such number words originated. Positioning finger-counting as the earliest form of materially structured counting in Mesopotamia was inferred from the way finger-counting appears to work in other societies globally. Further, the fingers and hand have not traditionally been classified as material structures for counting, though an extensive literature attests to their being a somatic basis for it. Archaeological evidence suggests there were early tallies, which take the

1. INTRODUCTION

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form of worked bones from the Epipaleolithic Levant.18 Not only is this time and place temporally, geographically, and culturally distinct from the later Mesopotamian societies who used tokens, examinations that can categorize notches as possibly quantificational in nature19 have not been performed on these artifacts, to the best of my knowledge. Tallies are the weakest link in my chain of inferences, though they fill what seems an otherwise inexplicable gap in the sequence of material devices, the transition off the body to one-dimensional material devices. The Neolithic clay tokens are fairly well known at this point, mainly through the extensive publications of SchmandtBesserat, as are their correspondences with later, unambiguous, numerical notations.20 However, much like fingers and hands, written signs have not been previously investigated as material structures in the elaboration of number. But given status as a material structure, they too fill an explanatory gap, the conceptual change historically described as the transition from concrete to abstract numbers. Whatever inferential holes mar the argument that fingers, tallies, tokens, and numerical notations form a material sequence elaborating initial number concepts into counting sequences and mathematics, however non-generalizable the sequence may be to societies other than Mesopotamia, the resultant framework nonetheless holds the potential to explain numerical prehistory cross-culturally. That is, material devices have long been recognized as representing and manipulating numbers.21 What is new about the present work is that it explains what those material structures do in number concepts—not just anchoring and stabilizing them,22 not only acting as proxies for their properties,23 but providing the very mechanism of elaboration. New devices for representing and manipulating numbers extend some of the capabilities provided by older devices, resolve some of their limitations, and inject new limitations that at some point may motivate the incorporation of even newer devices. This idea draws upon the work of ecological psychologist James Gibson,24 whose theory of material structures having exploitable properties (or affordances) was most illuminating and useful in this regard. Importantly, analysis of the properties, capabilities, and limitations of the devices used in Mesopotamian counting revealed an internal consistency to the sequence. 18 Coinman, ‘Worked Bone in the Levantine Upper Paleolithic: Rare Examples from the Wadi Al-Hasa, West-Central Jordan’; Reese, ‘On the Incised Cattle Scapulae from the East Mediterranean and Near East’. 19 d’Errico, ‘Memories out of Mind: The Archaeology of the Oldest Memory Systems’; ‘Microscopic and Statistical Criteria for the Identification of Prehistoric Systems of Notation’. 20 Friberg, ‘Preliterate Counting and Accounting in the Middle East: A Constructively Critical Review of Schmandt-Besserat’s Before Writing’; Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. 21 Ifrah, From One to Zero: A Universal History of Numbers; The Universal History of Numbers: From Prehistory to the Invention of the Computer; The Universal History of Computing: From the Abacus to the Quantum Computer. 22 Hutchins, ‘Material Anchors for Conceptual Blends’. 23 Frege, The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number. 24 Gibson, The Ecological Approach to Visual Perception; ‘The Theory of Affordances’.

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For this analysis, Malafouris’ Material Engagement Theory (MET)25 proved an apt framework. MET is able to compare cognitive states without implicitly favoring later states over earlier ones, addressing my concern with avoiding the potential pitfalls of cultural categorization and societal modes of thinking. Because MET envisions cognition as the interaction of brain, body, and world, it let me consider the respective contributions of, and interactions among, the psychological, behavioral, and material dimensions of numbers. The framework has aspects that are debatable, if not controversial. For example, MET views cognition as both extended (cognition includes materiality as a constitutive element) and enactive (cognition is the interaction of brain, body, and materiality). It’s a good question whether investigating the role of materiality in numerical cognition requires that materiality be constitutive of cognition, rather than causally linked to it. I was skeptical on this exact point when I first encountered Malafouris’ work because, as one of my professors once expressed it, causal linkage is easy to demonstrate, constitutivity much harder. Now after several years of learning to see cognition from a MET perspective, I have become a thorough convert. Readers of the present work will have to decide their level of commitment for themselves, of course. Here it will suffice to note that redrawing the boundaries of cognition to include materiality, as Malafouris invites us to do,26 raises the possibility of gaining new insights. I doubt the present work would have been possible without such redistricting. For example, as I investigated numerical notations as material structures, I also looked at how they differed from signs for nonnumerical language, and how both changed over time. After this analysis and much thought,27 I came to see reading as an unambiguous example of cognition that is both extended and enactive, because it is a cognitive state that does not—indeed, cannot— exist without engaging the material form that is writing. It is difficult to imagine what sort of thing reading could be, without engaging its material form. Once reading is admitted as example of extended and enacted cognition, what follows is the recognition that the material form has become increasingly effective at eliciting specific behaviors and psychological responses in its users. For other material forms, including those used for numerical representation and manipulation, what follows is not whether they are constitutive of cognition, but how. Mathematician Brian Rotman sees mathematical calculation by means of notations as an amalgam of thinking and scribbling,28 something Malafouris would describe as brain, body, and material forms being constitutively intertwined. And, as I explain in the book, the reason numerical notations differ from nonnumerical signs becomes explicable through Malafouris’ distinction between material and linguistic signs, and as a function of their different material prehistories. 25 Malafouris, ‘At the Potter’s Wheel: An Argument for Material Agency’; ‘The Cognitive Basis of Material Engagement: Where Brain, Body and Culture Conflate’; Malafouris and Renfrew, The Cognitive Life of Things: Recasting the Boundaries of the Mind. 26 Malafouris, How Things Shape the Mind: A Theory of Material Engagement. 27 Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’; ‘Thinking Materially: Cognition as Extended and Enacted’. 28 Rotman, Mathematics as Sign: Writing, Imagining, Counting.

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This work is only an initial step in understanding how peoples use material forms to realize concepts of numbers, and how those concepts change over time through the incorporation of new material forms. Accordingly, this work needs to be, and hopefully will be, subject to correction, revision, and expansion by other scholars interested in answering questions of numerical origins and change. One loose end that might be unraveled further is whether admitting materiality into numerical concepts sheds any new light on historical perspectives of what numbers are. That is, as concepts, numbers have properties that are unique and peculiar, like their intersubjective verifiability and our confidence in apprehending them.29 Certainly, peoples widely separated in space and time have come up with the same sequence of counting numbers, and given that sequence and a notion of divisibility, will converge on the same prime numbers. These properties have led thinkers since Plato to postulate that numbers are discovered, not invented, and as things that are discovered, must in some sense exist. If such properties truly originate in things like material semiosis and conceptual anchoring by the material forms used to represent and manipulate numbers, there may be some implications in this for mathematical realism. Along similar lines, the involvement of materiality in numerical concepts seems to at least partly refute the rather introspectionist idea that numbers are completely mental phenomena, the intuitionism of mathematician Luitzen Egbeurtus Jan Brouwer.30 Such extensions, however, must be worked out in future efforts, as they fall outside the scope of the present work. I hope this work will further two ambitions. First, I want to change the language we use to characterize the differences between and within number systems. Beyond their being the rather distasteful fruit of an outdated view on societal difference, the terms abstract and concrete are just insufficiently descriptive. Characterizing numbers instead by their content, organization, and structure will help us, I believe, gain traction on how peoples become numerate by using material structures, something with implications for research into how numbers originate, how they vary between cultures, and how they change over time, along with the much larger issue of how material forms inform the ongoing cognitive change of our species. Second, I want this work to build upon the past research that is its basis, not start a wholly new way of conceiving and researching Ancient Near Eastern numbers. I don’t merely wish to avoid the problem of there being two competing approaches, I also want to recognize that previous scholars have pointed out real phenomena, matters the current state of research into numerical cognition now enables us to understand from a different perspective. With these potential contributions to understanding numerical origins and elaboration, as well as the role of materiality in numerical cognition specifically and in human cognition and the development of complex cultural systems generally, I hope this work at last answers Professor Renfrew’s useful caveat and spur in a satisfactory and meaningful manner.

29 30

Frege, ‘The Thought: A Logical Inquiry’; Hersh, What Is Mathematics, Really? Brouwer, Brouwer’s Cambridge Lectures on Intuitionism.

CHAPTER 2. NUMBERS THROUGH A DIFFERENT LENS In analyzing numerical cognition, I used the Material Engagement Theory (MET) of cognitive archaeologist Lambros Malafouris as my theoretical framework. MET considers cognition as a system composed of brains, bodies, and materiality, and all of these things contribute to numerical cognition, albeit in different ways. Particularly important were the framework’s abilities to examine how materiality interacts with, and influences change in, behaviors and brains and how the cognitive system for numbers changes over time, as attested by material change. Historically, these things have been underexplored, not just in numerical cognition but in cognition generally. MET’s emphasis on material contribution to cognitive change was also appropriate, given my archaeological orientation to numerical cognition. MET has three commitments: First, cognition is extended and enactive. That is, cognition is a system that includes materiality as a constitutive component, and cognition is the interaction of brain, body, and materiality. Redrawing the boundaries to include materiality allowed me to think in terms of multigenerational interaction between brains, behaviors, and the material forms used in numbers, wherein each component influences change in the others. Change in behaviors and brains, then, followed from archaeological insight into change in material forms. Second, materiality has agency: Its properties influence our behaviors and psychological responses. This commitment suggests the materials forms used to represent and manipulate numbers influence how we use them and how we conceive numerical content, structure, and organization. Third and finally, materiality has a semiotic function (or a meaning potential), different from that of language, which makes it a material sign. Simply, things are meaningful in virtue of what they are and what we do with them. Such enactive signification is important in numerical cognition, where material meaning precedes language for numbers, and change in material form informs change in the way numbers are conceptualized. How these commitments pertain to my analysis of numerical cognition are examined in turn below.

COGNITION IS EXTENDED AND ENACTIVE The idea that the brain is not all there is to cognition is a philosophical perspective known as the extended mind hypothesis. It is important to clarify up front that seeing cognition as extended and enactive does not mean the brain is uninteresting, uninvolved, or unimportant. Rather, the brain is a vital part of a complex, dynamically interacting system that includes the body and material forms. While the brain has an un9

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ambiguous and important role, the body and materiality are also meaningful parts of the cognitive system. This status is seldom granted to materiality in particular, which tends to be viewed instead as merely ‘the stimulus that triggers or mediates some cognitive process’1 understood as neural activity in the brain. Seeing materiality as constitutive of cognition is not the view of cognitive psychology and neuroscience, which have only recently begun to recognize that the body contributes, though this recognition at the moment seems limited to the ways in which physiological development affects brain function and form. Seemingly, the mainstream of cognitive science has yet to realize that materiality has a role beyond stimulating momentary responses or occasioning neurological change such as that seen in acquiring skill—like hearing music can influence mood and playing a musical instrument is associated with increased gray matter in the brain’s motor and somatosensory regions. This focus is reasonable, given the theories and methods currently available and the need to limit scope and potential confounds in a way that achieves research goals. Some, perhaps many, or even most readers may reject the idea that materiality is constitutive of cognition, preferring to think of cognition as something the brain does, and calling any attempt to include materiality as part of it into question. I wasn’t a convert at first either. However, we can, as Malafouris suggests, redraw the boundaries of cognition at least momentarily to see what new insights this might generate. In the traditional model, cognition is neural activity in the brain, and brains are causally linked to the material world. A phenomenon like sound perception is explained like this: Something in the world disturbs the air, whose vibrations strike the eardrums, causing neurons in the brain’s auditory and association cortex to activate, with the result that the person hears a noise and identifies its likely direction and source. In this model, causality flows directly from a tree falling in the proverbial forest to someone hearing the sound it makes. And of course, there isn’t a sound if any of these elements are missing: Causal linkage requires the chain of causes and effects to be unbroken. Causal linkage in sound perception, however, does not readily convey the idea of the interactivity and bidirectional change that occurs between brains, behaviors, and materiality when someone plays the violin in an orchestra. Here another, more dynamically interacting model is called for, one that recognizes that as a species, we are uncommonly adapted for manipulating material forms in ways that involve massive amounts of feedback between what we experience, what the material form does, how we respond to what it does and how it responds to our response, and so on. Another example is reading, a cognitive state that does not—indeed, cannot—exist without our interacting with the material form that is writing. A third is pencil-and-paper calculation, which mathematician Brian Rotman has characterized as an ‘amalgam of two [inseparable] activities, thinking (imagining actions) and scribbling (making ideal marks)’. 2

1 Malafouris, ‘The Brain-Artefact Interface (BAI): A Challenge for Archaeology and Cultural Neuroscience’, p. 265. 2 Rotman, Mathematics as Sign: Writing, Imagining, Counting, p. 39.

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Such amalgams are what Malafouris calls the ‘constitutive intertwining of brains, bodies, things and cultural practices’.3 The traditional model characterizes proficiency in terms of ontogenetic change, wherein behavioral skill and neurological change are acquired through practice. Skill can also be reduced to change in the motor and sensory neurons involved in fine physical movements. However, even the traditional model recognizes that proficiency requires both a body and its engagement of material forms: No one produces music with the violin or becomes literate or good at long division without actually making actual movements with the necessary instrument. The traditional model also assumes the material forms themselves don’t change in any way, except in the trivial sense of being manipulated during their use and eventually wearing out. This view keeps agency firmly tied not just to the person but to the brain. When considered static and inert, material forms become simply the things we use, and the brain remains firmly at ‘the centre of the human world’.4 Yet none of these objects—not the violin, neither writing nor numbers—were invented ex nihilo as the forms they are today. They have not only changed from earlier forms, they have become highly adept at eliciting specific behavioral and psychological responses in their users, through the incremental change in their forms accumulated over generations of tinkering and adjustment. The extended, enactive view enables us to see materiality as a mechanism for creating and recreating behavioral and psychological change in societies and generations, not just individuals. It says something about how societies use materiality as repositories for knowledge and skill, how future generations draw upon these resources to instill behavioral and psychological change in themselves, and how this in turn enables them to tinker with and adjust the material forms they use. From an archaeological perspective, change in material form represents sustained behaviors and interactions with psychological processes. This opens a new window on how ancient peoples thought with their material forms, and how we think with ours as well, and how our interactive change with material forms continues to unfold. Using materiality to store our knowledge and skill is remarkable, in both a practical and an evolutionary sense. Materiality, broadly speaking, consists not just of material artifacts, but of the cultural practices and social institutions whereby we use them. Cumulatively, these allow us to decompose problems into series of smaller tasks, which are more easily solved because they are smaller. This effect is seen in numbers, where a complex task like multiplying 8912 by 7252 can be reduced to ‘a series of simpler problems beginning with 2 × 2’.5 Decomposing problems also makes it easier for multiple individuals to collaborate in solutions, opening up new possibilities for outcomes, in addition to the variation gained through ‘difference, localism, and choice’.6 The in3 Renfrew, Frith, and Malafouris, ‘Introduction. The Sapient Mind: Archaeology Meets Neuroscience’, p. 1936. 4 Renfrew and Malafouris, ‘Steps to a “Neuroarchaeology” of Mind’, p. 381. 5 Hodder, Entangled: An Archaeology of the Relationships between Humans and Things, p. 35. 6 Robson, Mathematics in Ancient Iraq: A Social History, p. xxii.

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volvement of multiple individuals also increases the potential that we might realize something novel, the two-heads-are-better-than-one effect writ large. Solutions can take the form of artifacts, which makes them available to other individuals and future generations; this opens up further opportunities to refine or apply artifacts to new uses, affording additional possibilities for change. When material culture enables future individuals to perform tasks and solve new problems, cognitive effort becomes distributed over space and time, the insight of cognitive anthropologist Edwin Hutchins.7 Using materiality to distribute cognitive effort is one of the major differences in how the human species uses materiality, relative to other species. Chimpanzees, our nearest living primate relatives, fish for termites. Adult chimpanzees, typically females, fashion termite-fishing probes from twigs, use them to extract termites from their mounds, eat the termites, and discard the twigs. Juveniles watch this performance, understand the goal of their mothers’ behavior, and are later able to fashion their own twig-tools to extract and eat termites. Nowhere in this sequence of making, using, discarding, observing, and emulating is there any sign of the tools themselves being retained to act as a point of departure for future effort. In contrast, we do not make and afterward discard a calculator every time we perform a mathematical task, just as our ancestors at some point saved their tally sticks. A calculator represents invention, retention, refinement, instruction, and learning in numbers, notations, algorithms, mathematical pedagogy, metallurgy, plastics, electronics, software, data storage, marketing, and transportation, a non-exhaustive list that demonstrates an amount of cognitive effort that no one individual or society could easily marshal. Someone using a calculator to perform a mathematical task is unlikely to be any of the people who invented the things that went into the device, which instantiates all sorts of knowledge in ways that help new individuals perform tasks and discover applications. Cumulatively, processes of invention, structuring, learning, use, refinement, extension, and conceptual distribution afford opportunities for devices, behaviors, and brains to change, especially over cultural spans of time. Artifactual designs are refined and extended to become more efficient, capable, and tailored to support new or increasingly specialized purposes. Modern devices instantiate, in a very meaningful sense, the ancestral cognitive activity this represents, not just material forms and the behaviors they influence, but the repurposed and novel brain functions and forms. This interactivity changes our brain function and form. Ontogenetic change occurs when individuals acquire performative skill with specific material forms, like learning to read trains the fusiform gyrus to recognize written characters, interact with the motor region that plans the specific movements of handwriting, and interact with the functions that produce and comprehend speech. Such behaviors, sustained over cultural spans of time, can repurpose brain functions. Reading, a thoroughly cultural function, leverages the ability of the fusiform gyrus to recognize objects, a function endowed by our evolutionary history. This repurposing of existing functions to new purposes is what psy-

7

Hutchins, Cognition in the Wild.

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chologist Stanislaus Dehaene calls neuronal recycling.8 Over even longer spans of time, new brain regions can emerge, like the regions of the human intraparietal sulcus specialized for detailed vision, neuroanatomy without a homologue in monkey intraparietal sulcus. These specialized regions help us wield tools with greater skill, representing a neuroanatomic change developed through millions of years of ancestral tool-use.9 Admittedly, the extended/enactive model leaves significant issues unresolved. If music, reading, and mathematical calculation are plausibly extended and enactive, cognitive states like reverie and dreams are less so. If music and reading admit specific material forms to specific cognitive states, things like tables and coffee cups seem to remain outside our cognition. Arguably, once violins, books, and algorithms are admitted to be constitutive of specific cognitive states, the question is no longer whether tables and coffee cups are part of cognition, but how. This is a decidedly difficult question to answer, since there are few criteria by which to judge. Philosopher Andy Clark proposed function as a criterion.10 When Otto writes himself a note to assist his memory, the note performs the same function as his remembering without material assistance. Since they are functionally indistinct, the standard by which organic mental recall is deemed cognitive while writing helpful notes is excluded can be questioned. Debate then centers around whether these forms of memory are truly equivalent, and the basis on which such external aids can be considered constitutive of cognition, rather than just causally linked to it. Perhaps function is not enough. Even among music, reading, and mathematics, there are differences in things like which body parts move and how much, the proportions of motor movements to sensory perception, the amount and types of knowledge recalled and applied, the degree to which the material form is incorporated into the body, the amount and predictability of movement in the material form, and how sustained the engagement of the material form seems to be. All these suggest potential criteria, with some very likely to be more pertinent than others, and the list is hardly an exhaustive one. I do not solve these issues in the present work, and indeed, such was not my ambition. I merely adopt the view that numerical cognition incorporates and encompasses material devices in a way that exceeds causal linkage. This was useful for examining the question of how we bridge the gap between the perceptual experience of quantity, which is shared with other species, and the conceptualization of number, which appears unique to humans. Simply, realizing explicit concepts of number requires the manipulation of materiality, which make numerical properties tangible and manipulable.11 Few species use their paws, hooves, or pads to explore and gain information

Dehaene and Cohen, ‘Cultural Recycling of Cortical Maps’. Orban et al., ‘Mapping the Parietal Cortex of Human and Non-Human Primates’. 10 Clark, Supersizing the Mind: Embodiment, Action, and Cognitive Extension; Clark and Chalmers, ‘The Extended Mind’. 11 Coolidge and Overmann, ‘Numerosity, Abstraction, and the Emergence of Symbolic Thinking’; Malafouris, ‘Grasping the Concept of Number: How Did the Sapient Mind Move beyond Approximation?’. 8 9

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about the world, and none to the extent humans do.12 We rearrange the world so that it makes different sense to us. This is especially true in numbers, where the hand integrates the mental and physical senses of quantity to facilitate our forming explicit concepts of number. Developmental psychologists may quibble with this statement on the grounds that if it were true, then training the hands would improve mathematical task performance. This does not appear to be the case, at least for the individuals with deficient calculating abilities with which it has been tried. Archaeology, however, generally demands that we think beyond the individual to consider groups as large as societies and temporalities that encompass lineages. Without the abilities to engage materiality through the hands and use the hand itself as a device for representing quantity, numbers might well have remained locked within the limits of our perceptual experience of quantity. More severely, were we unable to incorporate material structures into our cognitive system for numbers, we might not have formed explicit concepts of even the very small quantities salient to our perceptual experience, since the tactile manipulation of objects and reference sets appears necessary to the process from its onset. The constitutive intertwining of materiality and brains and bodies is thus vital, the very mechanism by which the perceptual experience of quantity becomes human numerical conceptualization.

MATERIALITY HAS AGENCY Agency is the capacity of an agent to act in the world. It is often conceptualized as something humans have, related to their abilities to form and pursue courses of action. In this regard, it is sometimes called free will, and it often means intentionality. I stipulate here an assumption that I think everyone but the hylozoist will accept: Material forms lack free will and the kind of intentionality we humans believe ourselves to have. However, agency can also be understood as a relational property, something that causes change within a system. Material artifacts, in changing human behavior, display agency: A speed bump insists drivers slow down or risk car damage and the attendant need for repairs; a stone, in fracturing unpredictably when being shaped, co-creates the actions of the person shaping it.13 Thus, an agent need not have free will, intentionality, or even be alive in order to act. Material agency is no mere passive repository for human intentionality. Natural stone has agency in producing (or not) a sharp, usable edge, making it suitable (or not) for use as a handaxe, affecting outcomes of activities like hunting and informing decisions about whether a group returns to the locations where it is found. A totem has agency in enforcing taboos, altering beliefs and behavior in those who acknowledge its power.14 The sextant has agency in enabling sailors to estimate latitude and longitude, improving their chances of safely reaching their desired destination.15 Though neither Gallagher, ‘The Enactive Hand’. Malafouris, How Things Shape the Mind: A Theory of Material Engagement; ‘Knapping Intentions and the Marks of the Mental’. 14 Gell, Art and Agency: An Anthropological Theory. 15 Hutchins, Cognition in the Wild. 12 13

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equivalent to, nor interchangeable with, human agency, material agency is nonetheless complementary and relationally intertwined with human agency to an extent that makes it difficult to separate the two in any meaningful sense. While materiality has agency in influencing behaviors and brains, the specific changes it is able to effect is a function of its properties, or affordances. As theorized by ecological psychologist James Gibson, affordances are the capabilities an environment offers to an agent, and they are found in the interaction between an agent’s abilities and its environment, rather than being properties of either:16 The affordances of the environment are what it offers the animal, what it provides or furnishes, either for good or ill. The verb to afford is found in the dictionary, but the noun affordance is not. I have made it up. I mean by it something that refers to both the environment and the animal in a way that no existing term does. It implies the complementarity of the animal and the environment.17

Affordances are inherently related to an agent’s abilities, both physical and mental; if a species lacks the ability to exploit a particular affordance, that affordance is not one for that species. A monkey has little use for a calculator beyond mashing its buttons to display the lights or using it as a projectile. Clearly, such employments also fall within the human behavioral repertoire, as do using calculators as paperweights, decorations, and dust collectors. But most of us use calculators to perform mathematical tasks, and indeed, some of us cannot perform the calculations needed for the task at hand without the affordance the calculator provides. The calculator also shows that affordances are not just features of the natural environment: They can be encoded as material artifacts. When affordances take the form of material artifacts, their capabilities influence behaviors. The sextant is a navigation instrument that represents navigational knowledge accumulated, applied, and improved by generations of individuals, as well as the algorithms implicit in, and automated through, the artifact’s design. Like a calculator does for numerical knowledge, the sextant instantiates the ‘kinds of knowledge that would be exceedingly difficult to represent mentally’.18 Such artifacts make it likely that current tasks and problems will be approached in certain ways. Navigation will be performed by means of the sextant, mathematics problems will be solved with a calculator, rather than using or inventing some other method that does not make use of the knowledge and capabilities they encode and provide. What do material agency and affordances look like in numerical cognition? First, we must consider how we perceive quantity, the so-called number sense or numerosity. It’s an ability that has been found, to the best of my knowledge of the literature, in all the species in which it has been investigated: not just primates and mammals, but birds, reptiles, amphibians, fish, and perhaps insects as well. Generally, only very small quantities are perceptually salient—one and two, usually three, and occasionally four. QuanGibson, The Ecological Approach to Visual Perception; ‘The Theory of Affordances’. Gibson, The Ecological Approach to Visual Perception, p. 127. 18 Hutchins, Cognition in the Wild, p. 96. 16 17

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tities above about four are simply many, and this is true regardless of whether or not we are able to count. The ability to appreciate very small quantities is called subitization, and it is consistent with, and thought to be limited by, the resources of cognitive processes like attention, object tracking, and memory.19 In a classic experiment, William Jevons, a 19th-century economist, came up with a simple yet effective way to measure subitization. He threw small, random quantities of dark-colored beans onto a light-colored surface, judged their quantity with a glance, and then checked his impression of their quantity against their actual number.20 His accuracy was high whenever the quantities were subitizable; it dropped quickly and considerably whenever the number of beans was higher than about four. Above the subitizing range, we discern differences in the quantity of groups, an ability known as magnitude appreciation. To be perceptible, the difference must be above a threshold of noticeability, the neurally based Weber–Fechner constant that also governs perceptual modalities such as temperature, weight, sound volume, light brightness, and color.21 That is, when two objects are held, one in each hand, their temperatures or weights must differ sufficiently for the difference to be perceptible; when they aren’t sufficiently different, the person holding them will not be able to judge that one is cooler or lighter, the other warmer or heavier. Similarly, when we look at two groups of objects, their quantity difference must be above the noticeability threshold for us to be able to tell them apart as bigger and smaller. Naturally, other characteristics affect such judgments, like the size of the objects and their location in relation to one another. In experimental protocols, researchers control for such characteristics in order to focus on numerical discrimination. So, when we are confronted with more than about four of anything, we do not see their quantity, and again, this is independent of whether we can count. If we want to know their quantity, we must count them, if we can, and if we cannot, we must rearrange the objects in some fashion. We tend to rearrange them into groups whose quantity falls within the subitizing range. This practice is found in early forms of written numbers, where the number seven might be written as a group of three vertical wedges above a group of four. We might also create groups whose quantity is accessible and manipulable. Such is the case in the familiar tally marks grouped by fives, as four vertical marks crossed by one horizontal or diagonal. This grouping leverages our ability to count things with decimal numbers. We might also put the objects into an arrangement whereby their quantity can be compared to that of something whose quantity is already 19 Burr, Turi, and Anobile, ‘Subitizing but Not Estimation of Numerosity Requires Attentional Resources’; Carey, The Origin of Concepts; Ester et al., ‘Neural Measures Reveal a Fixed Item Limit in Subitizing’; Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’. 20 Jevons, ‘The Power of Numerical Discrimination’. 21 Dehaene, ‘The Neural Basis of the Weber–Fechner Law: A Logarithmic Mental Number Line’; Fechner, Elemente Der Psychophysik; Masin, ‘The (Weber’s) Law That Never Was’; Nieder and Miller, ‘Coding of Cognitive Magnitude: Compressed Scaling of Numerical Information in the Primate Prefrontal Cortex’; Weber, De Pulsu, Resorptione, Auditu et Tactu.

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known, a strategy common in cultures whose numbers are relatively unelaborated. Because its quantity influences our behavior, materiality can be said to have agency. Material agency is also found in the affordances of the material forms used for counting, which shape or constrain our behaviors. Consider the hand as a device for representing and manipulating quantity. Most individuals in all human societies have hands, and using the fingers for counting appears cross-culturally universal. This prevalence is independent of how elaborate a particular number system becomes, as it is found across the full spectrum, from number systems that are emerging to those that have become highly elaborate. Finger-counting and the range of numerical elaboration where it occurs reflect the neurological integration of the abilities to perceive quantity and know the fingers. These functions are associated with adjacent regions of the parietal lobe, the intraparietal sulcus and angular gyrus. Their neurological integration in numbers is demonstrated by the functional impairments that occur when the angular gyrus is damaged: finger agnosia, an impaired ability to know the fingers; an impaired ability to count with the fingers; and acalculia, an impaired ability to understand and process numbers.22 In addition to this integration, the fingers have quantity themselves as five discrete digits, bridging our perceptual experience of quantity with material objects that have quantity, moreover material objects we know in a particularly intimate way. Their neurological integration and ready availability predispose us toward using the fingers for counting, and by instantiating quantity, they make the concept of it tangible, accessible, and communicable. Fingers have other influences on how we conceive quantity. When number systems start to emerge, numbers are often represented by pointing out some aspect of the environment with the requisite quantity. At the high end of the subitizing range, such designations may be approximate, rather than discrete. For the Gooniyandi, an Australian aboriginal people of Western Australia, the word for three ‘does not precisely designate “three” (but indicates rather “a few”)’.23 The Gooniyandi also use the fingers on one hand to indicate the number five. Collecting such numbers onto the hand will influence them toward properties implicit in the device itself, as well as how we tend to use it. Discreteness emerges because it is difficult for a number that means about three to remain loosely defined when it is represented by a finger in a sequence, where it necessarily becomes defined against the representations of two and four on adjacent fingers. Sequentiality emerges because it is easier to access the quantities being represented when we use a device the same way every time, which reduces the demand on psychological resources like attention and working memory. And linearity emerges because it is implicit to forms like the hand and the tally, and this artifactual linearity reinforce the 22 Grabner et al., ‘Fact Learning in Complex Arithmetic and Figural-Spatial Tasks: The Role of the Angular Gyrus and Its Relation to Mathematical Competence’; ‘To Retrieve or to Calculate? Left Angular Gyrus Mediates the Retrieval of Arithmetic Facts during Problem Solving’; Roux et al., ‘Writing, Calculating, and Finger Recognition in the Region of the Angular Gyrus: A Cortical Stimulation Study of Gerstmann Syndrome’; Seghier, ‘The Angular Gyrus: Multiple Functions and Multiple Subdivisions’. 23 McGregor, A Functional Grammar of Gooniyandi, p. 149.

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linearity inherent to the topographic organization of our primary sensory and motor cortex24 and the serial production of sound in speech. Further insights about material agency are found when the fingers are compared to beads as a device for representing and manipulating quantity. Like hands and fingercounting, beads are widely cross-culturally prevalent. They span the same range of numerical elaboration: Devices like the rosary require neither concepts of, nor words for, numbers in their use, while beads have been used to count into the tens and hundreds of thousands by peoples as widely separated by geography, culture, and language as the Pomo of Northern California25 and the Igbo of Southeastern Nigeria.26 And beads have a very long prehistory in Africa and the Near East, with many such archaeological finds dated to 100,000 years ago or earlier. But while most societies make and wear beads, very few count with them. This may be the case because fingers and beads offer different affordances for counting. Beads are not neurologically integrated with our ability to perceive quantity the way fingers are, so their use in counting does not have a similar prepotency. Where the hands are used in most human activity, beads must be repurposed to counting, a change that requires altering habits and concepts related to their use as ornaments. Not all the differences in their affordances are negative. Where fingers are generally, but not invariably, limited to low numbers, beads have an expanded potential for higher numbers, and where the fingers have a limited persistence in instantiating quantity, beads have a durability that potentially spans generations. How arithmetical operations emerge is influenced by material agency through the affordances of the material form(s) used for counting. The types of material forms found in emerging number systems generally share the ability to accumulate. It is easy to extend another finger or add another notch to a tally, knot to a string, stone to a pile, tear in a leaf, or mark on the ground, at least up to the capacity of the material form: the number of fingers, the surface for notches, the length of the string. Such accumulation implies relations of more, less, same, dissimilar, and perhaps one more, as well as the potential for them to become explicit. Fewer of these technologies have as ready a potential for subtraction. Notches are difficult to fill in, tears impossible to repair, though knots can be untied or cut off, stones removed, and marks on the ground erased. Fingers, despite their universality in counting, are seemingly not a factor in developing subtraction; perhaps the embodied experience of quantity does not disappear when an extended finger is flexed in the same way it does when a stone is removed from a pile. Given the affordances for accumulation (all) and subtraction (some), it cannot be a coincidence that, as a rule, addition emerges first and subtraction develops later. Regularity, manipulability, and integrity are other characteristics where material affordances differ. Notches on sticks and knots in strings instantiate the kind of stable24 Harvey et al., ‘Topographic Representation of Numerosity in the Human Parietal Cortex’; Patel, Michael, and Snyder, ‘Topographic Organization in the Brain: Searching for General Principles’. 25 Barrett, Material Aspects of Pomo Culture. 26 Zaslavsky, Africa Counts: Number and Pattern in African Cultures.

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order linearity that fingers afford, a similarity perhaps a factor in why such technologies tend to follow finger-counting. By comparison, stones and other noncontiguous objects lack stable-order linearity as a property implicit to their form, but their individual elements can be manipulated in ways that notches, knots, leaf-tears, and fingers cannot. Greater manipulability is a likely factor in extending the recognition of numerical relations beyond one more. Four pebbles, for example, can be decomposed and rearranged as three and one, one and three, or two and two, groupings with the potential to illuminate not only the quantity relations involved but their relative equality, as two and two balance in a way that three and one do not. A notched tally, however, maintains the integrity of the instantiated quantity in a way that loose, manipulable counters do not, unless they are somehow contained. As these examples illustrate, potential outcomes for numerical elaboration may be enabled or constrained by the affordances offered by particular material forms. Affordance differences may also be potential factors in the incorporation of new material forms, since the need to count higher or represent longer than what the fingers afford may motivate the use of a material form that can do one or both of these things. Material agency and the variability admitted through affordance differences may be as significant in influencing outcomes in the cognitive system for numbers as the various decisions, applications, needs, and values of the social dimension.

MATERIAL SIGNS ARE ENACTIVE MET’s third tenet is enactive signification, the idea that things are meaningful because of what they are and what we do with them. Understanding how materiality contributes to conceptual meaning starts with the difference between a word for an object and the object itself. A word is ephemeral: Once spoken, it disappears, though we are likely to retain it in memory long enough to connect it to other words, yielding phrases and sentences and discourse. Strings of words occur in a linear sequence, a function of how we produce the sounds of speech. And the meanings of words are conventional, arbitrary associations between sounds and meanings that must be learned. In contrast, a material object persists, a quality that helps it anchor and stabilize our concepts of it and associated properties like quantity and weight. An object has no inherent start or stop or directionality to it, though we may converge on a habitual method of handling it, as this will reduce the demand on our attention and improve the accessibility of the information it represents. And an object instantiates its meaning: It means what it is. This fundamental difference in how objects and words mean can be illustrated by thinking about what might happen if we asked children either to arrange objects or create names for them. Conceivably, those asked to arrange the objects might form ideas about their quantity, ideas prompted by the interaction of sorting, ordering, and rearranging. Notions like divisibility, for example, might emerge when groups of objects are split into smaller groups that provide opportunities for noticing the quantity of their elements. This is because things have quantity, quantity is meaningful to our number sense, and that meaning becomes clearer to us when we put things into different configurations. The situation is different when it comes to language. The strings of sounds that make words and phrases are arbitrary—meaningful only when, and because, everyone uses them the same way. Associations between things and words are learned, not inherent in the objects themselves. Rearranging the objects will not alter

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the situation. Thus, children asked to create names for the objects are likely to invent names based on what they know about words through their exposure to language. They will put the sounds of their language together in ways that resemble the words they know. Simply, objects cannot suggest names in language the way they can suggest concepts like four and relations like two plus two. For properties like weight, the associated concept or symbol ‘cannot exist without the substance, and the material reality of the substance precedes the symbolic role’.27 This reflects ‘an inherent link between the physical and material and the symbolic. … Units of weight are indeed conceptual, but they would be unthinkable without the experience of the physical reality of weight’.28 Renfrew doesn’t mention quantity or concepts of number, but I believe his point holds for them just as well. This distinction of physical and conceptual reality evokes a thought experiment offered by philosopher Frank Jackson, in which Mary, a scientist, knows everything there is to know about the color red but has never seen it herself. The question is whether she learns something new once she finally does see it. Jackson argued that the experience of properties like color cannot easily be reduced to physical matters, and that when Mary finally sees red, she will gain ‘a certain representational or imaginative ability’ that she previously lacked.29 This is representationalism, the idea that brains internally recreate a reality external to them. Renfrew is not affirming that brains represent; rather, he is observing that concepts of weight and quantity depend on our having experience of things with these properties. We gain this experience by interacting with objects, not just once, but constantly, dynamically. This is what philosopher Andy Clark once dubbed the 007 Principle: The world tells us only what we need to know, when we need to know it. 30 What the principle does is substantially reduce the amount of representing the brain needs to do, by shifting the burden of representing back onto the world itself, as its own, best model. The result is a brain that does very little representing internally but instead interacts with the world to gain its information. In non-representationalist cognition, we interact with the world to get the information we need. This is perception, not as passive observation, wherein brains merely wait, receive, and process external stimuli delivered to them by the world, but as active process. If we fix our gaze for even a short period of time, we lose visual information. Most of us never realize how much movement vision actually requires—movements that direct and focus the eyes, changes to the orientations and angles of the head, transits of the body through space. It is through this kind of active engagement that meaning is brought forth. We understand things like weight and number because we interact with things that have them, not just once but continually, and our interaction is the mechanism whereby such concepts are brought forth.

Renfrew, ‘Towards a Theory of Material Engagement’, p. 23. Renfrew, ‘Commodification and Institution in Group-Oriented and Individualizing Societies’, p. 98. 29 Jackson, ‘What Mary Didn’t Know’, p. 294. 30 Clark, Being There: Putting Brain, Body, and World Together Again. 27 28

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21

Meaning becomes ‘the product of a process of “conceptual integration” between material and conceptual domains’.31 The idea is that concepts are blends of what the brain does, what material structures are, and our active engagement of them, wherein material forms act to anchor and stabilize our concepts of them.32 For numbers, this is a particularly illuminating way to envision them as concepts, because number concepts are rooted both in object quantity and our perception of it. Representing and manipulating quantity by means of objects that have it makes our perceptual intuitions tangible, discrete, and explicit.33 Object properties and relations then become proxies for numerical properties and relations,34 allowing us to conceptualize numbers as having properties like linearity and manipulability that they gain from the material forms we use to represent and manipulate them. And our perception of quantity continues to underpin and inform our numerical concepts, independent of how elaborated they might ultimately become in terms of quantity, relations between quantities, and the patterns formed by relations of quantity. Quantity, quantity relations, and relational patterns are also independent of whatever we happen to call them in language, once we get to the point where we call them something in language. The perception of quantity is not acquired through language, since it is demonstrated by species without language. In the human species, the alinguistic experience of quantity generates concepts that are typically expressed through iconic or indexical means. An icon conveys by resembling, and an index points in some fashion to what it means. Iconic means of expressing quantity include recreating it with fingers or objects. Sound can also be used iconically, as the Mundurukú, an indigenous people of Amazonian Brazil, use words with two, three, and four syllables to convey, respectively, the numbers two, three, and four.35 Indexical means of conveying quantity involve pointing to an object that has it, either with a physical gesture or with words. The Abipones, who once inhabited the lowland Gran Chaco region in Argentina, invoked ‘the [four] fingers of an emu’ and a ‘skin spotted with five different colours’ to represent the numbers four and five.36 Quite often, the numbers five and ten are the first to emerge from the undifferentiated many, the term used across languages and cultures for non-subitizable quantities. These numbers are represented by the hands, enough to make ten predominant as an organizing base. Using the hands combines the iconic use of an object with indexical gesture, indexical phrases like ‘as many as the fingers on my hand(s)’, or both. Once we express numbers in language, there are striking differences between their linguistic and material forms. One of the most interesting is seen when written numbers, or numerals, are compared to written non-numerical language. Numerals do not Malafouris, How Things Shape the Mind, p. 90. Hutchins, ‘Material Anchors for Conceptual Blends’. 33 Frege, The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number; Malafouris, ‘Grasping the Concept of Number’. 34 Hutchins, ‘Material Anchors for Conceptual Blends’. 35 Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’. 36 Dobrizhoffer, An Account of the Abipones, an Equestrian People of Paraguay, Vol. 2, p. 168. 31 32

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encode the phonetic information needed to identify the associated words of a particular language. This is the difference between Roman numerals and their names in Latin, and it is the quality that makes numerals semasiographic, writing that is meaningful without the intercession of language. Thus, the familiar Hindu–Arabic numerals 1, 2, and 3 are simultaneously one, two, three in English, uno, dos, tres in Spanish, ek, do, teen in Hindi, yî, èr, sân in Chinese, and waahid, athnaan, thlaaatha in Arabic. Indeed, phonetic encoding would make them much less usable as numbers, because the addition would degrade their ability to make plain the numerical, spatial, topological, and geometric relations that allow us to access and manipulate numerical relations and patterns. This is the difference between multiplying a column of figures and performing the same information on a list of words. Because they are semasiographic, numerals retain a conceptual and physical manipulability37 and ability to represent relations of spatiality and transformational invariancy38 similar to that found in other, non-written forms used to represent and manipulate numbers. These properties also have no counterpart in written nonnumerical language, except perhaps for things like anagrams and crossword puzzles. However, even for anagrams, transformations do not have the potential to generate concepts of relations and patterns like numbers do. Phonetic independence makes numerals more similar to other material forms used to represent and manipulate number than it does written non-numerical language. That is, while three fingers and three cuneiform wedges share the property of three-ness and mean three because they instantiate three, no such inherent similarity relates the word head to a picture of one. A picture resembles, certainly, but it does not instantiate what it resembles, making it somewhat ambiguous regarding the word it is intended to represent. Conversely, what instantiates may not resemble, but nonetheless is what it means. The lack of phonetic encoding also means that numbers tend to be the first, and perhaps the only, things identified in otherwise unknown languages and scripts, like the Linear A of ancient Minoa, whose associated language remains unidentified, and the Elamite script, used in what today is southwestern Iran between the 3rd and 1st millennia BCE. It might not be possible to tell what the associated words were, but their meaning as numbers is intelligible. This issue will resurface in discussing the numerical vocabulary that would have been associated with written numerals and earlier technologies for counting in Mesopotamia. I will take the position that because they instantiate, material forms for counting attest to the existence of an associated vocabulary, even if the material forms themselves do not encode the phonetic information that identifies the actual words. Instantiation of quantity without phonetic encoding may explain why numerals, like other material forms of number, are similar across significant linguistic, cultural, and temporal differences: One, two, and three vertical or horizontal strokes are signs for subitizable numbers in many systems of numerical notation. Such forms also tend 37 Landy, ‘Toward a Physics of Equations’; Landy and Goldstone, ‘Formal Notations Are Diagrams: Evidence from a Production Task’. 38 Larkin and Simon, ‘Why a Diagram Is (Sometimes) Worth Ten Thousand Words’; Sfard and Linchevski, ‘The Gains and the Pitfalls of Reification? The Case of Algebra’.

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to be conserved. Today’s Hindu–Arabic 1, 2, and 3 are essentially the same straight lines used since the beginning of writing,39 despite their transmission across thousands of years and adoption by multiple cultures and languages. The same process yielded radical alteration in non-numerical writing, as scripts were adapted to different lexicons and phonemic inventories. This difference in semiotic function means that numerals remain inherently material, despite being written, despite their written form becoming less depictive over time, and despite acquiring attributes that are mentally understood or behavioral rather than explicitly represented. This last point is important, for material meaning does not exhaust what materials mean. We see this in numerals today. While the material forms of the symbols explicitly represent quantity, other essential information, like numerical base, is implicit and must be learned and added mentally.40 Similarly, in ancient Greece, dot matrices helped mathematicians visualize the relations within numbers. A number like four could be represented as two rows, each containing two dots, helping to instantiate the notion of two squared being four.41 In Neolithic Mesopotamia, small geometric shapes made of clay known today as tokens were used to represent numbers. Like the Greek dots, four tokens could be arranged as two rows of two. While this representation is nearly identical to the later Greek matrix, it represents a distinct notion of what numbers are. The Greeks thought of numbers as entities in a relational system, while Neolithic Mesopotamian peoples were more likely to have thought of them collections with the same quantity as the commodity being counted. The difference between the ancient Greek and Neolithic Mesopotamian concepts of numbers is a function of their material history, the totality and sequence of all the material forms used to represent and manipulate them. Numerical properties reflect the properties of material devices used to represent and manipulate them. Numbers may continue to have these properties, across and despite being represented or manipulated on new material forms that lack the properties in question. This persistence is the result of behaviors, habits, expectations, and knowledge, matters acquired through interactions with older material forms that influence how newer forms are used. As the structure of older material forms persists in way newer ones are used, conceptual content, organization, and structure may change to incorporate aspects of both. Numbers become linear and ordered in a stable fashion when, and because, they are represented on devices like fingers and tallies that have an implicit linearity and stable order to their form. Linearity and stable order persist in the way tokens and numerical notations are used, though these forms are inherently neither linear nor ordered in themselves. That is, in addition to retaining the linearity and stable order of older forms, numbers add

39 Branner, ‘China: Writing System’; Chrisomalis, Numerical Notation: A Comparative History; Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer; Nissen, ‘The Archaic Texts from Uruk’; Martzloff, A History of Chinese Mathematics; Tompack, A Comparative Semitic Lexicon of the Phoenician and Punic Languages. 40 Zhang and Norman, ‘A Representational Analysis of Numeration Systems’. 41 Klein, Greek Mathematical Thought and the Origin of Algebra.

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manipulability when they are represented on manipulable forms like tokens or an abacus.

NUMBERS THROUGH A NEW LENS Obviously, ancient brains can’t be studied directly. However, cognitive science provides insight into numeracy, and the archaeological record provides material forms like tokens and numerals that attest behaviors like counting and imply psychological processes like the number sense and categorization. Change in material form, then, implies change in behaviors and brains. This analysis requires two things: a material record with the duration and extent to show granular change in material form, and a cognitive state understood well enough that change in material form can suggest change in behaviors and brains. Now, this is not causal linkage. It’s just analytically useful to talk in terms of the interaction of brain, body, and world yielding measurable change across the different components of the cognitive system. When we redraw the boundaries of numerical cognition to include materiality and its engagement, we get a different picture of what numbers are as concepts and how they are realized from the perceptual experience of quantity. This picture puts material forms front and center. This doesn’t mean the brain isn’t crucial or special in an evolutionary sense, or that the behavioral and material components contribute the same things as the brain. But once we have included the material and behavioral components, we have a model that lets us consider cognitive change in societies over cultural spans of time, matters intractable to current psychological theories and methods. The model helps us understand how societies composed of average people invent things like mathematics in the first place. It may also give us some insight into how we ourselves think through material forms, and the relationship between conceptual thought and material forms more generally. This analysis encourages us to think about materiality as something that accumulates the cognitive effort of past generations and recreates the associated behaviors and psychological responses in present and future ones. Material forms change as this occurs, becoming better able to elicit specific behavioral and psychological responses from their users. Here materiality also acts as a collaborative medium, one that influences group behaviors and brains toward common change. Group tools create communities of specialists like scribes and mathematicians, people able and identified for using particular tools. Group tool use also affects tool form. Though human behaviors and brains have much in common, people differ in their psychological, physiological, and behavioral attributes. Most of this individual variability cancels itself out—the highs counteract the lows—keeping tool forms synchronized to average user capabilities, while allowing those forms to change. Synchronization in turn distributes the average to new users, influencing commonality and cohesion in the behaviors and psychological processing of the members of the social group. But these matters require more space than the current chapter allows to discuss them. And a more detailed discussion of what numbers are as concepts is required, along with what the ethnographic literature shows about how number concepts emerge. I address these in the chapters that follow.

CHAPTER 3. WHAT’S A NUMBER, REALLY? Surely we know what numbers are, since we use them every day—right? But arguably, few people have pondered the old and deeply philosophical question of what Aristotle might have called their essence. Asking a roomful of college undergraduates tends to elicit statements like, ‘A number is how many of something there are’. Asked to define the number six, a student once told me, ‘It’s the number between five and seven’, which another swiftly corrected to, ‘It’s what you get when you add four plus two’. Interestingly, these statements are not just all correct, they also reflect different aspects of what numbers are. The first statement notes their cardinality, or how many there are of things in a group. The second locates six in an ordinal sequence, numbers placed in order of their increasing size: It’s five, six, seven, and never six, five, seven or any of the other potential orders. The last statement specifies one of the additive combinations that yield six, others being three plus three and five plus one. Cardinality, ordinality, and additivity reflect elements of a more formal definition, like the one that specifies number as ‘a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication’. 1 Succession says that new numbers can always be formed by adding one to any other number, implicitly expecting that the newly formed numbers, each next of which is larger, follow those they were formed from, which are smaller. This orders numbers by increasing size, or magnitude, a quality of ‘enormous importance’ because it gives numbers ‘most of their mathematical properties’.2 Magnitude ordering doesn’t specify the size of the intervals in the sequence, so each new number is merely larger than the one it follows. Succession, however, stipulates the intervals as one, making each new number one larger than its predecessor. Their order, and the size of the intervals in their ordering, form the basis for the relations between numbers. Their relations let us manipulate numbers by addition, subtraction, multiplication, and division, as was the case earlier with four and two making six because six is two more than four and four is two less than six. Such relations, as the philosopher Plato observed almost 2500 years ago, are what make numbers numbers: They are not as meaningful individually as they are in relation

1 2

Merriam-Webster, definition 1c2 of ‘Number’. Russell, Introduction to Mathematical Philosophy, p. 29.

25

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to one another, just as the relations between notes are what make music music and relations between sounds comprise speech. Such relations, and the operations used to manipulate them, also make each number unique. One is the only number that always yields whatever number it multiplies or divides, including itself, even if these operations are performed an infinite number of times; this peculiarity excludes one from being considered a prime number. Two is the only even prime, as all other even numbers are divisible by two and hence are not prime. Three, the first odd prime, is also the only number that sums all previous positive integers, being one plus two. Among just these first three numbers, there are many relations: three is more than one or two, and one and two are both less than three; one is the same as one and not the same as two or three; one is also two minus one and three minus two; two is both one plus one and three minus one; and three is one plus two, two plus one, and one plus one plus one. Adding the number four to the sequence increases the amount of potential relations, as well as the likelihood of discovering them and the patterns they contain. Prime numbers are one such pattern; others are odd and even numbers, counting by fives and tens, and the Fibonacci series, where each number sums the two preceding it. Mathematics has been called the science of detecting such patterns: ‘numerical patterns, patterns of shape, patterns of motion, patterns of behavior, and so on’.3 But eliminate all the formulas, consolidate the component elements, and the definition that begins to emerge is this: Numbers are discrete quantities ordered by increasing magnitude and related to each other in specifiable ways, with operations like addition that manipulate them by means of their relations. Not all numbers have all these properties. That is, number is no monolithic entity. While we might tend to think that what numbers are to us today is what numbers are to everyone, in every place and for all of time, this is what mathematician Brian Rotman calls a backward-looking determinism.4 A monolithic view of number risks more than seeing some numbers as incomplete: it also fails to notice how our familiar Western numbers have changed over the past several thousand years. Along the way, things that initially were not numbers have joined their ranks, including zero,5 one,6 two,7 fractions, and negative and imaginary numbers. Numbers have become applicable to anything and everything: ‘For number applies itself to men, angels, actions, thoughts, every thing that either doth exist or can be imagined’.8 They can be represented by, and manipulated on, a variety of forms, like fingers, tallies, abaci, notations, and calculators. Seeming matters of common sense, like the idea that one plus one makes two, have been

3

Devlin, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe,

p. 3. Rotman, Mathematics as Sign: Writing, Imagining, Counting. Rotman, Signifying Nothing: The Semiotics of Zero. 6 Klein, Greek Mathematical Thought and the Origin of Algebra. 7 Evans, ‘From Abacus to Algorism: Theory and Practice in Medieval Arithmetic’. 8 Locke, An Essay Concerning Human Understanding, Book 2, Chapter XVI, Sect. 1. 4 5

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questioned on the grounds that we do not know that this will be true for all of time and under all conditions, unless we can somehow prove that it will.9 While Western numbers have changed over the course of their history, a similar variability is seen when numbers are considered across cultures. In Papua New Guinea, traditional Oksapmin numbers are counted from one through twenty-seven using specified positions on the body, an expanded type of finger-counting. In terms of their quantities, Oksapmin numbers are identical to those of the West, and they are a size-ordered sequence, with the relations implicit to ordinality: more, less, same, different. When psychologist Geoffrey Saxe asked his Oksapmin respondents to perform a subtraction task, they invented an impressively complex double-enumeration strategy: Subtracting 7 from 16, for example, involved creating ‘internal correspondences within the body system, using one series of body parts, in this case the thumb (1) through forearm (7) (the subtrahend), to keep track of the subtraction from 16 (the minuend)’. 10 Beyond their sequencing, ordinal numbers lack the idea that 7 and 16 differ by 9. This restricts operations like addition and subtraction to only those ways permitted by ordinal relations—counting up and counting down. And along the basin of the Amazon River are several cultural groups with even fewer numbers than the Oksapmin’s twenty-seven, including the Mundurukú and Pirahã in Brazil. The Mundurukú count from one to four,11 while the Pirahã have been famously described as having no numbers whatsoever.12 The differences go beyond the extent of the numerical range, as Mundurukú numbers are not merely fewer: The higher ones tend to be approximate, rather than discrete. That is, their word for four might as often mean about four than exactly four. Certainly, there’s nothing in this changeability to contradict the idea the numbers are real, a notion long associated with Plato. He said numbers were universals, or abstract, repeatable properties that exist as ideal forms, independently of the physical things that exemplify and instantiate them in the world or the minds that apprehend them. Universals are held to exist in some way still rather poorly defined, despite the millennia of philosophical energy expended on them since Plato’s day. Also ill-defined is the way human minds might come into contact with Platonic universals in the first place, given that their apparent immateriality makes them invisible and intangible. Nonetheless, as concepts go, numbers have some undeniably peculiar properties, giving Plato’s view of them a significant boost. One is that the very same sequence of counting numbers is formulated, time and again, by different peoples with different cultures and circumstances. Another is that implicit to those numbers are the very same relations and patterns within the relations. People will not only find the same re9 Peano, Arithmetices Principia: Nova Methodo Exposita; Whitehead and Russell, Principia Mathematica. 10 Saxe, Cultural Development of Mathematical Ideas, p. 86. 11 Pica et al., ‘Exact and Approximate Arithmetic in an Amazonian Indigene Group’; Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’. 12 Everett, ‘Cultural Constraints on Grammar and Cognition in Pirahã: Another Look at the Design Features of Human Language’; Frank et al., ‘Number as a Cognitive Technology: Evidence from Pirahã Language and Cognition’.

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lations and patterns, they will always get the same result when multiplying or dividing this by that. They will also agree, with unusual conviction, that the answer is right, the intersubjective verifiability noted by mathematician Gottlob Frege.13 Given these qualities, the idea that we discover numbers seems correct. As things that are discovered do seem to exist, discoverability gives numbers an existence external to the human mind. In this conceptualization of number, any change to what we think numbers are merely represents our discovering new aspects that were there all along, or our correcting some mistaken notion because we have come to know better. Essentially, to the realist, numbers don’t change, but our familiarity and understanding improve over time. Some hint we might not be entirely satisfied with the realist view lies in the continuing debate. Were realism a satisfactory answer to the questions of what numbers are and where we get them, one might think any debate would have long since ended. In polar opposition to the idea that numbers are real entities existing externally to the human mind is the idea they are purely mental constructs. Here ‘mathematics becomes the study of mental constructions of a certain type’.14 Taking this view, mathematician Luitzen Egbertus Jan Brouwer proposed what is called intuitionism, the idea that number concepts are realized by ‘the consciousness of self in time’.15 Brouwer ‘roughly sketched’ his view of how we intuit numbers through a somewhat mystical ‘inner experience’16: twoity; twoity stored and preserved aseptically by memory; twoity giving rise to the conception of invariable unity; twoity and unity giving rise to the conception of unity plus unity; threeity as twoity plus unity, and the sequence of natural numbers; mathematical systems conceived in such a way that a unity is a mathematical system and that two mathematical systems, stored and aseptically preserved by memory, apart from each other, can be added; etc.17

Brouwer was dissatisfied with the realist answer, but he also wanted to establish a secure foundation for mathematical truth. Foundationalism generally means starting with incontrovertible knowledge; conclusions drawn from that knowledge are then assured of being sound by being derived from true premises with a valid method. Mathematical reasoning, in Brouwer’s view, was valid, which meant its conclusions would only be sound if they were anchored to bedrock truth. He chose numbers for this foundation, in part because they have been recognized as the basis of mathematics for so long, their position as such is definitional: Mathematical systems are ‘obtained by extension

Frege, ‘The Thought: A Logical Inquiry’. Chomsky, The Generative Enterprise Revisited: Discussions with Riny Huybregts, Henk van Riemsdijk, Naoki Fukui and Mihoko Zushi, p. 43. 15 Brouwer, Brouwer’s Cambridge Lectures on Intuitionism. 16 Brouwer, p. 90. 17 Brouwer, p. 90. 13 14

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of or analogy with the natural number system’,18 the familiar numbers in the sequence one, two, three used for counting in everyday speech. He was also aware that previous foundationalist attempts had failed, sometimes spectacularly so, like the attempt by Gottlob Frege, and later Bertrand Russell and Alfred North Whitehead, to derive all of mathematics from logic and sets. Brouwer’s choice of a starting point was therefore the key to his own success. In Brouwer’s conceptualization, mathematical structures were derived from numbers, and numbers in turn arose in the mind as a function of consciousness in time. His notions that numbers originated as mental content and that this would give them the requisite foundational solidity came from ‘the old metaphysical view that individual consciousness is the one and only source of knowledge’.19 This view of mental content has a long and distinguished history. In the 17th century, Descartes used it to establish a foundation for knowledge generally, proving he existed and so did the world and God and what he knew of both through his access to his own mental content: He perceived his thoughts, so he must exist and so must the world; he couldn’t be deceived about what he perceived because God existed and was good. Access to mental content, however, is not nearly as infallible as Descartes wanted to believe. Much, if not most, of our mental content is unconscious, and hence, inaccessible to introspection. The conscious part is altered when we introspect, degrading the accuracy of what we think we find. We may misunderstand, misinterpret, or misreport what we think we find, making our reports of mental content unreliable. Reporting is also highly individual, and there are few criteria for reporting mental content in terms of categories, category assignments, and priorities. At most, introspection provides a limited insight into mental content; ultimately, it lacks the validity and reliability needed to justify claims about knowing what goes on in the mind, let alone the grander claims proving all knowledge and existence or basing mathematical certainty and truth upon it. To be sure, Brouwer might well have been mistaken about his own mental content in relation to numbers, in being predisposed to think of them as arising, easily and fully formed, in the mind, as they must have in his. He was born and enculturated into a society whose numbers were both highly elaborated and deeply interwoven into the cultural fabric. Beyond an impressive mathematical talent, his avocation as a Cambridge mathematician meant he spent much of his time thinking about numbers, giving him an insight and familiarity well beyond what is typical for most. Brouwer was also a product of the time. Cross-cultural variability in number systems was known, but during Brouwer’s time and even toward the end of the 20th century was attributed to 19thcentury notions about different modes of thinking. Concrete thinking was what socalled primitive societies did, while Brouwer’s own, presumably advanced, enlightened, rational, and scientific culture thought abstractly. Societies with fewer numbers than his own were, baldly stated, considered less than human, a characterization that has con-

18 19

Merriam-Webster, definition 1c2 to ‘Number’. Ferreirós, ‘The Crisis in the Foundations of Mathematics’, pp. 148–149.

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tinued to permeate scholarly discourse into uncomfortably recent times, if in disguised or weakened form.20 Brouwer might have been inclined to defend his view of numerical origins, as well. If numbers were real, or constructed like he thought mathematics were, rather than introspectively intuited as he claimed, the critical foundation of his mathematical system would have been kicked right out from underneath him. He was also naïve about numerical cognition, a field of study that has added much, during the fifty-plus years since Brouwer’s death, to what we know about what brains do and how peoples and language behave in numbers. This evidence that he didn’t and couldn’t have known is addressed in Chapter 4.

AN ARCHAEOLOGY OF NUMBER AND A ROLE FOR MATERIAL FORMS Earlier I said that numbers are discrete quantities like one, two, and three that are ordered by their increasing size, related to each other in numerically specifiable ways, and manipulated by means of their relations. This definition decomposes our everyday Western notion of number into several constitutive elements that can then be traced to the kinds of material forms used with numbers—which is, after all, the type of evidence that archaeologists are best known for. Certainly, historians of mathematics like Georges Ifrah21 have noted the use of different material devices throughout the ages. These devices have been used to perform two basic functions in numbers: representing and manipulating. Archaeological evidence of devices used for these purposes has the potential to provide insight into numerical prehistory at a depth of time that things like reconstructing the number words in a proto-language cannot touch. While linguistic techniques currently reach back a maximum of 10 or 15 thousand years, some artifacts possibly used for numbers are tens of thousands of years older, placing them in the European Paleolithic and African Middle Stone Age. Admittedly, deciding whether a 47,000-year-old notched bone or 87,000-year-old strung beads were used to count something, rather than meaning merely that someone once whittled or wore a necklace, is challenging. This has motivated archaeologists like Francesco d’Errico22 to offer methods and criteria intended to remove at least some of the ambiguity from determining use in artifactual forms like tallies. Beyond the advantage of the archaeological time depth and the challenge of archaeological interpretation are the cognitive implications of the devices themselves. Why do we use material devices to represent and manipulate numbers, and what happens when we do? Previous accounts have tended to answer the first question along the lines that physical devices act as external memory storage, aiding our cognition by 20 Verran, ‘Aboriginal Australian Mathematics: Disparate Mathematics of Land Ownership’, p. 292. 21 Ifrah, From One to Zero: A Universal History of Numbers; The Universal History of Computing: From the Abacus to the Quantum Computer; The Universal History of Numbers: From Prehistory to the Invention of the Computer. 22 d’Errico, ‘Memories out of Mind: The Archaeology of the Oldest Memory Systems’; ‘Microscopic and Statistical Criteria for the Identification of Prehistoric Systems of Notation’.

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offloading our mental content. Memory being stored externally means the brain has an idea, which when marked on a physical form makes the material represent the idea. The brain’s job becomes easier because the device assists its representation of mental content. Representational accuracy and persistence improve because devices don’t misremember or forget the way brains do. This account of devices in numbers assumes that brains do the work of representing, that material devices are passive repositories for the brain’s mental content, and that causality flows from brain to device. But—if brains don’t represent, if materiality is active, if causality flows not just from inside to out but from outside to in as well—the conclusions should accordingly change. Previous accounts have not really addressed the second at all, the question of what happens when we use material devices for numerical representation and manipulation. Partly this is because artifacts are assumed to be passive depositories for mental content with no causal backflow that can affect the brain. Partly too it’s a consequence of the archaeological record, where the oldest artifacts tend to be single devices isolated in time and space; this means there is nothing to compare them to that would raise questions like why these forms and why this sequence. When devices are not isolated, they have been considered not as sequences but as contemporaries, excluding opportunities to ask why one material form might follow another and how multiple forms might affect behaviors or the concepts themselves. And partly it’s a result of thinking that things like tallies and tokens are physically material, and hence, appropriately subject to archaeological theories and methods, while things like fingers and notations are biological or symbolic and accordingly, less amenable to analysis as part of material sequences. Placing such things in material sequences is problematic for other reasons, like the data being from sources archaeologists don’t usually consider, like language, or the data spanning times and places to an extent that sequences are implied, rather than attested unambiguously. For this analysis of how materiality contributes to what numbers are and how we get them, I adopted the strategy that all data are fair game, provided there was a rational basis for including them. That basis might be inferred from what we see other people do in numbers, in terms of devices, behaviors, or language. It might be genetic data linking ancient peoples in Mesopotamia to even earlier ones in the Levant, times and geographic regions loosely affiliated by the known transmission of agriculture and the observation that numbers, more often than not, are transmitted through cultural contact and descent. The result is imperfect, of course, as any inferential argument must be. But my conclusions provide at least an initial insight into why material forms are used to represent and manipulate numbers, and the effects these forms have on the concepts. Simply, numerical content, structure, and organization can be traced to properties of the material forms used in sequence to represent and manipulate numbers. Some of these effects occur because we incorporate material forms into our cognitive system for numbers, others through material agency and meaning. And as an initial starting point, the theory will be subject to criticism, correction, refinement, and extension, through my own subsequent research and, hopefully, that of others. Implicit to any notion of what numbers are, the one I use not excepted, is a history—and prehistory—in which countless individuals interacting with material forms have realized and refined numerical concepts, relations, and operations, which material forms make available to other individuals and future generations. This does not mean

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material forms exhaust the numerical concepts they realize and transmit, since some of the associated knowledge, habits, and norms transfer between individuals and generations in the form of behaviors, through language, or both. We see this today in numerical notations, signs in which attributes like place value are implicit and learned, not explicitly represented. Ultimately, what material forms mean is both what they are and what we have been taught they are and how we should use them. Also implicit to my understanding how numbers emerge and become elaborated is the idea that our contemporary, Western notion of number differs from earlier, nonWestern notions, including those of the Ancient Near East. There is a warning in this as well, not to impose our ideas of what numbers are on ancient peoples. What we want is a situated notion of number rooted in the things we know that ancient peoples would have shared with us—human physiological characteristics, like our five-fingered hands and toes; psychological capacities such as the number sense and language; and our tendency to use material devices for numbers—but which also expects the concepts themselves will recognizable and related but not identical. Indeed, there is a long and detailed history of how numbers have changed, though these tend to start when people begin writing things down. While writing has rather striking effects on numbers as concepts, it arrives relatively late in the sequence of material devices used for numbers, and so it sheds only a partial light on the origins question. Illumination must be sought in other evidence: the archaeological record and observations of how contemporary devices are used with numbers, especially by peoples whose numbers are relatively unelaborated.

TWO USEFUL THEORETICAL CONSTRUCTS Decomposing numbers into concepts, relations, and operations is merely a first step in understanding how numbers are realized and elaborated as discrete quantities, counting sequences, arithmetic, and mathematics. The next step requires an important distinction be introduced: the difference between number and cardinality. For any group of objects, cardinality is the number that specifies their amount. In a culture that already has numbers, cardinality is discerned by counting the members of the group sequentially from one until there are no more to be counted. The number associated with the final object stands for the number of objects in the group, its cardinality. Though number can be understood as cardinality and cardinality can be discerned by counting with numbers, the two are nonetheless distinct. As Russell noted, ‘A particular number is not identical with any collection of terms having that number: the number 3 is not identical with the trio consisting of Brown, Jones, and Robinson. The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections, namely, those that have that number’.23 That is, a number is a concept of the cardinality that sets of objects share. At this point, Russell’s logical types (Table 3.1) may be helpful. Admittedly, Russell did not intend his typology to answer the question of numerical origins, but rather, 23

Russell, Introduction to Mathematical Philosophy, pp. 11–12.

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to resolve some thorny issues in the theory of infinite sets. There are four logical types in Russell’s typology. Individuals, objects, and entities (Type 1) are distinguished from collections or sets of things (Type 2). Sets of things have cardinality, as for example, the two-ness of two fingers. Readers familiar with set theory will know that single individuals also have cardinality, as does an absence of individuals; these are, respectively, sets with one and no members. The point need not detain us, as the distinction would presumably not be familiar to peoples realizing their very first numbers. A number (Type 3) is cardinality shared by sets of Type 2, as in Russell’s comment about the number three being something all trios had in common. Sets of shared cardinalities, or numbers, are counting sequences (Type 4). Type 1 2

Entity Objects Sets of Type 1

Description Individuals, objects, or entities Cardinality of a set of objects

3

Sets of Type 2

Cardinality shared by sets, a number

4

Sets of Type 3

A set of Type 3 entities

Example a pair of objects two-ness (cardinality of a pair) two (two-ness shared by two or more pairs) the natural numbers (of which two is a member)

Table 3.1. Russell’s logical types. Though the types were not formulated to address the question of numerical origins, they suggest a framework for understanding how the perceptual experience of quantity might yield concepts of numbers. Based on a similar presentation of Russell’s types in Soames, Philosophical Analysis in the Twentieth Century. Vol. 1, The Dawn of Analysis, p. 155.

I am strongly drawn to Russell’s typology because it differentiates matters I think need to be teased apart to investigate what numbers are as concepts and how we get them: our perceptual experience of quantity in objects (Type 1) and group of objects (Type 2), our recognizing that quantity is shared between groups of objects, recognitions that are concepts of number (Type 3), and our forming sequences of numbers (Type 4). Another reason the typology seems like a good place to start answering the origins question is that it doesn’t assume numbers already exist. This is an important point, since a method of invention shouldn’t presuppose what it invents, a problem in accounts of numerical origins that assume concepts preexist in the mind or only exist once they’ve assumed linguistic form. In addition, Russell’s idea that numbers are shared cardinalities implies a mechanism for realizing number concepts and counting sequences, which is perceiving quantity and representing and manipulating it with sets of objects—material forms. For cultures without numbers, the first two Types are perceivable as small quantities and quantity differentials, the third requires rearranging materials to compare their quantity, and the fourth involves collecting those shared quantities onto a coherent material form to represent them as a sequence. This mechanism also simplifies the concepts that must initially be realized, since ‘it is simpler logi-

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cally to find out whether two collections have the same number of terms than it is to define what that number is’.24 Comparing quantity in sets of objects to realize concepts of number, and representing numbers with material forms, are the kinds of bootstraps and scaffolds MET predicts, ones that start with interactivity between psychological processes, behaviors, and material structures and produce conceptual content. Moreover, they are consistent with what peoples do when their language has relatively few terms for numbers. Behaviors that compare sets, like pairing and one-to-one correspondence, are well documented in the ethnographic literature. In pairing, two objects are physically grouped, enabling an appreciation of their cardinality or two-ness. Pairing pairs, like a pair of objects and a pair of fingers, plausibly opportunizes a recognition that pairs of any kind of object share two-ness, and this is the number two. Pairing involves subitizable quantities, either two as the number shared by pairs or one as the number of a single contrasted with that of a pair. This is not just how we perceive quantity, it also reflects the first words for numbers to emerge across languages, either two and one or one and two. In oneto-one correspondence, objects in two sets are iteratively matched until none remain unmatched and none have been matched more than once. This too plausibly creates opportunities to recognize that sets share cardinality, a number. Behaviors like pairing and one-to-one correspondence have tended to be dismissed as mere compensation strategies—what peoples do when, and because, they don’t have numbers. This characterization entirely misses the point: Such behaviors are the very process by which number concepts are realized. They are material engagements, the moments where brains connect with ‘the material affordances of … bodies [and] the agency of material culture and innovation’.25 Almost certainly, the logical types do not accurately depict what goes on in the brain, in terms of either psychological functioning or neural activity. Nor do they likely provide a realistic picture of mental content, a caveat meant to evoke my earlier observations on the pitfalls of Brouwer’s introspectionism. Their hierarchy suggests the involvement of multiple levels, and this in turn might imply concepts becoming ‘more abstract’ as they attain succeeding levels, matters that of course are not established beyond a reasonable doubt. On the contrary, it may be the case that once a concept changes, the understanding it engenders has priority over other, previous, understandings.26 A number understood through its relations with other numbers (Type 4) differs conceptually from one realized by comparing sets that share cardinality (Type 3). This does not mean the former is more abstract than the latter, but rather, that both, concepts with an equivalently abstract nature, have different content, with one displacing the other in our familiar way of thinking about numbers. It is equally uncertain that transitions from one level to another are discrete. Numeral classifiers, words that connect numbers to enumerated objects based on object properties like shape, suggests there may be intermediate Types interposed between those in Russell’s typology. These Russell, Introduction to Mathematical Philosophy, p. 15. Malafouris, How Things Shape the Mind: A Theory of Material Engagement, p. 110. 26 Brandt, ‘Language, Domains, and Blending’. 24 25

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drawbacks, however, are true of many such constructs, which nonetheless remain helpful in forming research questions and organizing and interpreting analytical results. A bigger drawback to Russell’s typology is its assumption that concepts like numbers are mental phenomena, distinct from the physical world and generated by, and contained within, the brain. As I mentioned in Chapter 2, representationalism in its strong form assumes brains reproduce the external world internally. Representationalism is also implicit to both Platonic realism and Brouwer’s intuitionism, realism because mental representation is perhaps the only way we might apprehend entities whose existence is otherwise immaterial and imperceptible to us, intuitionism because it assumes numbers originate as mental content. Exactly how much of the external world the brain actually reproduces is an open question.27 It probably doesn’t reproduce nearly as much as we might think. For one thing, reproducing the external world internally would require massive storage and enormous computational power, and the result would be slow and inefficient—hardly an evolutionary advantage28 or one that explains the preeminent adaptation of the human species, our use of material forms in cognition. Phenomena like change blindness also challenge the idea that brains recreate the world internally. A typical protocol for demonstrating change blindness involves a person who is part of the experiment asking another, the test subject, for directions. People carrying a large object then walk through the conversation. During the brief interruption, the person asking directions is replaced by another confederate, who differs from the first in appearance and dress but resumes the conversation as if nothing had disrupted it. Test subjects often fail to notice the confederate is not the same person who initiated the conversation. If strong representationalism were true, our internal representation of the world should be less vulnerable to such momentary change, even when it is less substantial. While representationalism in its strong form is likely incorrect, it is probably also not the case that there are no representations whatsoever, the strong form of antirepresentationalism. Off-line states like daydreams, reverie, and ‘reasoning about absent, non-existent, or counterfactual states of affairs’29 challenge strong antirepresentationalism, as do certain phenomena in numbers. One is the so-called mental number line, a language-independent but culturally influenced way of visualizing numerical magnitude. The mental number line is tested by asking participants to locate the position of a given number on an unmarked physical line. Children at an early stage of numerical acquisition, and individuals of all ages from societies with few numbers, tend to place the numbers logarithmically, bunching them together toward the ends of the line. Older individuals from societies with elaborated numbers tend to place numbers linearly, disposing them more evenly across the line. While not everyone agrees the mental number line construct is valid,30 the developmental change demonstrated by 27 Clark, Being There: Putting Brain, Body, and World Together Again; Supersizing the Mind: Embodiment, Action, and Cognitive Extension; Hutto, ‘Radically Enactive Cognition in Our Grasp’. 28 Clark, Being There. 29 Clark and Toribio, ‘Doing without Representing?’ p. 419. 30 Núñez, ‘No Innate Number Line in the Human Brain’.

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children from numerate cultures does suggest that external matters—things like cultural and linguistic exposure, education, reading direction, and material devices—influence how we conceive numbers, which sounds a lot like some internal representation is taking place. Like off-line states, the mental number line becomes more difficult to explain if strong anti-representationalism were true. If representations are neither all nor none, they logically inhabit the middle ground of some, without necessarily being fixed in any part of the intermediate spectrum. It’s also quite possible that the amount of representing we do varies with the activity of any particular moment: Off-line forms of cognition like daydreaming and mathematical visualization likely involve more representations than do skilled physical activities requiring our focused attention on sensation and movement, like making pottery.31 None of this precludes the very likely possibility that we have multiple, simultaneous engagements, wherein some of the concurrent activity is more representational than others, the kind of ‘dense structural coupling’ thought to exist between brain, body, and world in enactivist cognitive models.32 Evidence and better insight into what brains actually do will ultimately decide. Representation seems to explain what we seem to experience when we think, especially with numbers and mathematics. When we read and write elaborated material forms like notations and equations, we engage them visually and manually, and we visualize the associated concepts mentally, or in the mind’s eye. Mathematicians have long connected this mental visualization with material forms, recognizing the latter as something that supports it.33 Mathematical educator James Kaput saw representational acts as involving two components: mental activity, which ‘organizes and manages the flow of experience’, and a representational system, some kind of ‘concrete, external’ artifact, a category he broadly specified as including material devices, diagrams, notational symbols, and language.34 Kaput thought it particularly important to describe the structural features of mathematical representations in a systematic way, as well as how such structures interacted with each other.35 A more detailed model of mental representation offered by cognitive scientists Gilles Fauconnier and Mark Turner attempted to do just that: describe how the structure of representational inputs might interact to produce novel structure. They called their model conceptual blending. It had four interacting mental spaces: two input spaces, sometimes differentiated as source and target; an optional generic space that captured higher-level structure shared by the two inputs; and a blended space where these inputs all came together. They envisioned the blend to inherit some of its structure from the input spaces, with the interaction of the various

Malafouris, ‘At the Potter’s Wheel: An Argument for Material Agency’. Malafouris, How Things Shape the Mind, p. 99. 33 Dreyfus, ‘Advanced Mathematical Thinking Processes’, p. 31. 34 Kaput, ‘PME XI Algebra Papers: A Representational Framework’, p. 347. 35 Kaput, p. 354. 31 32

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inputs having the potential to produce novel structure as well.36 In the conceptual blending of numbers and two-dimensional space, the former acquired the latter’s angles, magnitudes, and coordinates, giving rise to emergent structure in the form of operations performed by means of these properties.37 The idea that such blending occurs wholly within the brain is a strongly representationalist take on conceptual formation and change, which I reject accordingly, and so might the mathematician who believes material structures support his mentally representing. Positioning material forms as essential to representing, cognitive anthropologist Edwin Hutchins offered a tripartite update to the conceptual blending model, a version of which appears as Fig. 3.1. It combined Fauconnier and Turner’s four mental spaces into a single mental input, which interacted with a material domain, with both projecting to an enactive third space, the blend. As in the original model, the blend inherits some of its structure from the inputs, and it has the potential for emergent structure. In this conception of how blending works, material forms act to anchor and stabilize representations, and their structural properties have the potential to inform structure in the resultant concepts.38 Here whatever amount of representation is going on not only involves material structures but is partly constituted by them. Conceptual properties are informed by our using the material forms involved in the blends, and conceptual permutations are created through their manipulations. This goes beyond Renfrew’s observation that concepts like units of weight require the physical experience of weight39 by suggesting material forms play not just an initiatory but a continuing role in our concepts. Again, such theoretical proposals may bear little resemblance to what actually goes on in the brain. No matter; Hutchins’ view of conceptual blending is still a useful way to think about how brains, behaviors, and material forms might interact to produce new concepts with novel structure, and it is nicely non-representationalist. Moreover, it has potential for modeling several phenomena found in numbers. One is that numbers have structural properties like linearity and manipulability that resemble properties of the material forms used to represent and manipulate them, and the model provides a mechanism for material structure to influence and inform conceptual structure. Another is that material forms like numerical notations imply rather than explicitly state information like place value, essentially combining external and internal representations.40 This is something the model expresses by having both material and mental inputs pro-

36 Fauconnier, Mappings in Thought and Language, pp. 149–151; Fauconnier and Turner, ‘Conceptual Integration Networks’. 37 Fauconnier and Turner, The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities, p. 243. 38 Hutchins, ‘Material Anchors for Conceptual Blends’; Malafouris, How Things Shape the Mind. 39 Renfrew, ‘Commodification and Institution in Group-Oriented and Individualizing Societies’; ‘Towards a Theory of Material Engagement’. 40 Zhang and Norman, ‘A Representational Analysis of Numeration Systems’.

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ject to an enactive space, where material forms are seen both for what they are in themselves and for what we have learned them to mean. When multiple material devices inhabit the material input space, similar to the multiple components inhabiting its mental counterpart, the model provides a mechanism for phenomena like contrastive value and persistent structure. Material forms can differ in how they represent and manipulate numbers, contrasts of content, structure, and organization with the potential to illuminate principles of numerical organization revealed when the various inputs interact, a key aspect of the model. Similarly, exposure to numerical differences through language gains entry to conceptual blending, either through the mental input space, for those who believe language to be a function of the brain, or through the material input space, for those who believe, as James Kaput apparently did, that language functions more like a material form. The model also provides a mechanism for material structure to persist, across and despite the adoption and use of new material devices that lack particular structure as an inherent property of form. This mechanism is the knowledge, expectations, and behaviors acquired with older material forms, which predispose people to use newer forms with the same behaviors used with older ones, even if newer forms have the potential to be used in other ways. Anchoring conceptual blending in numbers with material forms has another advantage: Its involvement of materiality opens up the manuovisual engagement of material forms as a locus of conceptual generation. Again, this is consistent with how mathematicians think they think, visualizing forms and conceiving relations by considering and rearranging material forms like equations, diagrams, and pictures. Moreover, concepts are generated not just by interacting with such material forms, but often by interacting with them in the absence of language. High-level mathematical thinking involves activity in many parts of the brain, but mostly regions that are not implicated in language, implying that such thinking is inherently alinguistic. 41 Complex mathematical ideas can be understood without language, as the dozens of proofs in Roger Nelsen’s Proofs without Words series42 demonstrate by stimulating mathematical thinking with visual clues alone. Language may even get in the way of such visualization. This has inspired some mathematicians to try to exclude language from mathematical visualization altogether, favoring material forms as communicating more directly with our visualizing abilities and seeing this mechanism as having greater generational value in mathematical conceptualization.43 The alinguistic use of material forms is also consistent with how initial number concepts emerge, as the ethnographic literature documents the use of hands, gestures, and material forms to instantiate quantity or point to things that instantiate it, well in advance of words being available to express the quantities in question. Obviously, language can be used to point out instantiating objects as well, a func41 Amalric and Dehaene, ‘Origins of the Brain Networks for Advanced Mathematics in Expert Mathematicians’. 42 Nelsen, Proofs without Words: Exercises in Visual Thinking; Proofs without Words II: More Exercises in Visual Thinking; Proofs without Words III: Further Exercises in Visual Thinking. 43 Silver, ‘The New Language of Mathematics’.

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tion I discuss in Chapter 6, but it is typically not the first method used for indicating quantity. I don’t want to exclude language so completely, for several reasons. One is that there is an undeniable role—several, actually—for language in numbers, especially for written language, particularly in mathematical elaboration, something I’ll discuss in conjunction with numerical notations and the invention of writing in the Ancient Near East. Another is that language is a valuable source of information on the material forms that have been used for numbers, illuminating aspects of their material prehistory. This is because material forms influence numerical structure, and thus, the structure of number words. An example is organizing base, where using the fingers and sometimes the toes as a material structure for representing numbers underlies the decimalization of number words.

Fig. 3.1. Conceptual blending with a material anchor. Blending combines inputs from mental and material spaces in an enactive third space, the blend. The lines between the mental, material, and enactive spaces represent ‘selective perception and projection’ prioritized and directed by processes like attention.44 Projection, ‘a non-representational conceptual mechanism’, connects ‘the supposedly internal and external domains’ in a ‘dense structural coupling’.45 Projection establishes connections of ‘identity, analogy, similarity, causality, change, time, intentionality, space, role, and part-whole, and in some cases also of representation’ between phenomenal mental and physical domains.46 Projection is trig-

Hutchins, ‘Material Anchors for Conceptual Blends’, p. 1561. Malafouris, How Things Shape the Mind, p. 99. 46 Malafouris, p. 100. 44 45

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THE MATERIAL ORIGIN OF NUMBERS gered and supported by external material forms47 that interact with, and influence, psychological and physiological capacities, knowledge and expectations, and behaviors. As they did in Fauconnier and Turner’s original model, conceptual blends inherit some of their structure from these inputs, and the interaction of the inputs has the potential to give rise to novel structure. Hutchins’ model differs from the original in including materiality and shifting the locus of blending activity from inside the brain to the interactivity between brain, body, and world. Adapted from Malafouris, How Things Shape the Mind: A Theory of Material Engagement, p. 101, Fig. 5.2.

In being organized by tens, fives, and twenties, amounts imposed by neither the number sense nor language, number words attest to counting with the hands and perhaps the feet. The number sense leaves its own mark on language, restricting words for emergent numbers to subitizable quantities and quantity differentials, and structuring grammatical number—how we distinguish singulars and plurals—to the same amounts when and if a language develops this feature. Though the information it provides is relative in its chronology, rather than absolute, language for numbers becomes a valuable source of information on the material prehistory of numbers, particularly for ancient peoples, whose behaviors and psychological capacities cannot otherwise be observed. Material forms have another effect on numbers as concepts, especially when several material forms inhabit the material input space, as they do in our highly elaborated Western numbers and once did in the Ancient Near Eastern material sequence. That is, we represent and manipulate numbers in a variety of material forms—symbols with different appearance and structure, like the familiar Hindu–Arabic notations and Roman numerals; bases other than ten, like binary, octal, and hexadecimal; devices like tallies, rosaries, clocks, and currency; and fingers—and our concept of number encompasses all these forms and more and shifts between them with such ease we barely notice that it does. This distribution represents numbers originating in, and becoming elaborated by means of, a sequence of material forms. During this process, numbers acquire new properties whenever new material forms are added to the sequence of devices, like the linearity and stable ordering influenced by the use of the fingers and the manipulability introduced through the use of tokens. Such properties can persist across change in material forms, as knowledge and habits acquired with older forms predispose us to use newer forms in similar ways. What this does, ultimately, is give numbers a set of properties originating from the use of different devices. Along with the novel structure that may emerge through conceptual blending, conceptual properties become different from the properties of any particular device used for representing and manipulating, as well as those of any set of devices. This makes concepts functionally independent of all the material forms that have informed their realization and elaboration, suggesting why the relation between material forms and human conceptual thought more generally is one of seeming independence.

47

Malafouris, How Things Shape the Mind, p. 102.

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We must do more than imagine multiple material devices inhabiting the material input space; we must also think about extending the model across multiple individuals at a time and across time. That is, the model has an implicit focus on individual activity in generating concepts. This focus should not exclude materiality’s important role in accumulating and distributing concepts between individuals and across time. This role is critical to concepts like numbers, not just because they are inherently social in nature but because they involve amounts of conceptual invention and elaboration beyond what any single individual or generation could realize on their own. The social dimension of conceptual blending is where knowledge and behaviors are transmitted to, and reproduced by, new individuals and generations, and social priorities and needs become able to influence conceptual elaboration. Here materiality acts as a collaborative medium for distributing new insights and behaviors to other individuals. This gives insights and behaviors opportunities to become incorporated into the collective body of social knowledge and be encoded as affordances of material devices, increasing the possibility they will persist across time. By acting as collaborative medium, materiality also enables multiple individuals and generations to contribute to conceptual elaboration, ultimately yielding highly elaborated conceptual systems like mathematics. Before leaving Russell’s logical types and Hutchins’ materially anchored conceptual blends for the psychological, ethnographic, and linguistic matters that inform how I interpret the archaeological record of the Ancient Near East, I want to address the potential relevance these theoretical constructs might have beyond the question of numerical origins. In Chapter 2, I introduced a thought experiment in which children given objects to sort, order, and rearrange might discover mathematical relations. This activity resembles a hands-on protocol used in education, wherein such exploration and revelation are thought to foster genuine mathematical insight while also being fun and memorable. Discovery depends on there being something to discover, and indeed, properties and relations of a numerical nature inhere in objects and inform our concepts of numerical content, structure, and organization. We can indeed rearrange groups of objects into smaller groups and notice that sometimes the groups we form are equal in number, and sometimes they are not, and perhaps this might lead us to speculate about matters of divisibility and remainder. Does tracing numerical properties to this origin and process of conceptual generation help us understand why numbers, concepts we create, can be used so often and so well to describe the world?48 Or why numbers are intersubjectively verifiable to a degree uncommon among concepts generally? Does the idea that numbers are produced by materially anchored blending, whether or not it involves language, give us better insight into how individuals apprehend and generate concepts, or how societies elaborate concepts over generations of time, perhaps in ways that let us improve the former or gain a measure of control over the latter? I won’t solve these issues in a few paragraphs, nor was solving them my goal. I merely note that locating numerical origins in an extended and enactive cognition

48

Hersh, What Is Mathematics, Really? p. 17.

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seems to have the potential for yielding insights into these matters, with the hope that someone might be inspired at some point to develop these ideas further.

CHAPTER 4. ASSEMBLING AN ELEPHANT, ONE BIT AT A TIME Investigating numerical origins brings to mind the old Buddhist parable about the blind men and the elephant: Everyone has solid evidence, and no one quite has the big picture. Assembling that picture requires sorting through the data generated by psychologists and neuroscientists, linguists, ethnographers, archaeologists, and of course, mathematicians, and then trying to see how everything might fit together within an explanatory framework. Cumulatively, these data make a lot of pieces for a large and complex unknown. Not all of these pieces may fit because not all of them are equally pertinent to the origins question. For example, a significant portion of the psychological literature focuses on how children typically acquire numbers as they mature, including characteristics of numerical language shown to help developmental acquisition. This, however, assumes a society in which numbers are already available to be learned, as well as a language with the specific helpful features. As such, the developmental data may be less immediate to the question of how numbers are realized in the first place, compared to data on how we perceive quantity. Our perceptual experience of quantity is where we’ll start.

NUMEROSITY, THE SENSE OF QUANTITY Psychologists and neuroscientists are concerned with brain function and form, things like the ability to perceive quantity, language as a cognitive function, maturational development in regard to acquiring and becoming able to reason with numbers, categorization and abstraction, the neuroanatomy for finger-counting, and the numerical effects of grasping objects. Of these, the function most immediate to the origins question is the so-called number sense or numerosity, the perceptual experience of quantity that underlies and informs how numbers are used. Numerosity is phylogenetically widespread, found in mammals, birds, reptiles, amphibians, fish, and perhaps even insects. This prevalence, despite vast differences of brain form and function among these various species, suggests the ability to discern more from less is evolutionarily advantageous, enabling individuals to choose more of things like food and shelter and less when it comes to competitors and predators. Since non-human species cannot respond to questions in language, demonstrating their ability to perceive quantity generally takes the form of presenting them with a scenario in which they must choose between larger and smaller, and then seeing how they respond. Fish can be given a choice of two groups of other fish to join, and when they reliably choose the larger group, they prefer 43

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what we might call safety in numbers.1 Salamanders, given a choice between more flies to eat and fewer, reliably choose more.2 As I mentioned in Chapter 2, our number sense is functionally divided into subitization, the ability to appreciate quantities up to three or four, and magnitude appreciation, the ability to appreciate bigger and smaller in quantities above about four. Subitization involves constraints on cognitive resources that limit the number of objects we can simultaneously attend to or keep track of, while magnitude appreciation involves differences in quantity large enough to be perceived. Fig. 4.1 demonstrates subitization and magnitude appreciation with non-symbolic quantity, the kind of quantity representation appreciated by non-human species under natural conditions, as well as humans whether or not we come from societies with numbers or happen to have learned them ourselves. Humans, of course, can also appreciate numbers in symbolic forms. In symbolic form of numerals, the quantities shown in the figure would be ‘3’ and ‘9’. For symbolic numbers, the ability to discriminate larger from smaller improves as the difference increases, a phenomenon known as the distance effect; when distance is constant, small numbers are more easily compared than large ones, the size effect.

Fig. 4.1. Subitization and magnitude appreciation. (Left) Subitization allows us to identify, rapidly and unambiguously, quantities from one to three and sometimes four. (Right) Magnitude appreciation lets us tell numerically bigger from smaller in groups, as long as the difference is above the threshold of noticeability governed by the Weber–Fechner constant. Without counting, the quantity on the right, which exceeds the limit on subitizing, is just many. Image created by the author.

Some non-human species, typically primates, have also possibly demonstrated an ability to understand symbolically encoded forms of number,3 though there is ongoing debate over whether and the extent to which species other than humans really comprehend symbols. For one thing, the conditions under which such behaviors emerge are highly artificial, involving not just captivity but extensive coaching by human tutors. Chimpanzees in particular also demonstrate a capacity for working memory that far exceeds that of humans.4 This leaves it unclear whether chimpanzees really compre1 Dadda et al., ‘Spontaneous Number Representation in Mosquitofish’; Piffer, Agrillo, and Hyde, ‘Small and Large Number Discrimination in Guppies’. 2 Krusche, Uller, and Dicke, ‘Quantity Discrimination in Salamanders’; Uller et al., ‘Salamanders (Plethodon cinereus) Go for More: Rudiments of Number in an Amphibian’. 3 Biro and Matsuzawa, ‘Use of Numerical Symbols by the Chimpanzee (Pan troglodytes): Cardinals, Ordinals, and the Introduction of Zero’. 4 Inoue and Matsuzawa, ‘Working Memory of Numerals in Chimpanzees’.

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hend numbers in symbolic form in the same way we do, or simply use their impressive memory to perform numerical tasks so well. What is clear is that even these primates, humanity’s most closely related living relatives, neither invent nor use numbers, symbolically encoded or not, in the wild. A more fundamental question is whether non-human species, when they respond to non-symbolic forms of quantity, really have a concept of number, a claim pervasive in the psychological and comparative literature. This question was taken up recently, and not without controversy, by cognitive psychologist Rafael Núñez.5 He proposed distinguishing the perceptual experience of quantity, which he called the quantical, from the conceptualization of number, the numerical. Núñez’ quantical–numerical distinction is foundational, dividing what we share with other species, the perceptual experience of quantity, from what only humans seem to do, culturally construct concepts of number. The two are often conflated, and not just terminologically as in the number sense. This is because our ability to perceive quantical information by means of our sense of number underlies and informs our ability to form and comprehend numerical information. This is demonstrated through a variety of psychological tests. In the mental number line (MNL), people mentally represent numbers as falling along a continuum. As was mentioned in Chapter 3, one test of the MNL involves having people place numbers along a continuum, where they demonstrate either a logarithmic or linear disposition, depending on their maturational development and cultural exposure to numbers. The MNL can also be tested by means of what is called the spatial-numerical association of response codes (SNARC), which purports to show how the number sense influences the way we envision and respond to numbers. The SNARC effect demonstrates a left– right preference for subitizable and non-subitizable numbers. People tend to respond faster when subitizable numbers are presented to the left visual field, non-subitizable numbers to the right. Conversely, reaction times are slower when small numbers are presented to the right visual field, large to the left. In addition to understanding how the number sense influences how we comprehend and manipulate numbers, the quantical seems like a good starting point for investigating how we get the numerical. This is because the number sense precedes not just the emergence of numbers, but the emergence of language as well in the evolutionary history of our lineage, the species Homo sapiens and ancestral species like H. erectus. The argument runs like this: The ability to appreciate quantity is so phylogenetically widespread among contemporary vertebrates—not just primates, not just mammals, but birds and reptiles and amphibians and fish—we can safely assume our remotest ancestors had it too. Because of the wide variance in brain form and function among all these different types of species, let’s limit our definition of the number sense to only what mammals have, so we are sure to be comparing brains with fairly similar functions and forms. Because the number sense is shared by primates with other mammals, it would have present in the earliest primate—an ancestral trait shared with other mammals, not one developed uniquely in primates. This means that all primate species, whether living or extinct, should have the number sense, and indeed, all living primate 5

Núñez, ‘Is There Really an Evolved Capacity for Number?’.

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species investigated to date have been found to have it. Primates emerged within the class of mammals between 50 and 60 million years ago, and no theory of language origins construes language as emerging at that depth of time. Thus, the phylogenetic distribution of the number sense shows our ancestors had it, and well before they developed language.

LANGUAGE AND NUMBERS Though preceded by numerosity, language was likely in place before—perhaps even long before—we began to create numbers. When you think language emerged depends on what you think language is. Linguist Noam Chomsky views language as a computational capacity so complex it could only have emerged once, and relatively recently at that, a mere 100,000 years ago.6 Other scholars, construing neurological and functional similarities between language and things like gesture,7 tool-use,8 and artistic depiction,9 have argued for a more gradual emergence over a much longer period of time. These accounts tend to place language origins much earlier, typically based on archaeological and fossil evidence demonstrating behavioral and anatomical features suggestive of language. Language in a rudimentary form, or proto-language, has been posited as emerging as early as 1.8 million years ago, in conjunction with the appearance of the Acheulean handaxe, a stone tool distinguished by its bilateral symmetry, or early Homo, species whose skulls suggest their brains had features associated with language.10 Whether you believe language to have emerged long ago or only recently, most of the possible archaeological evidence for numbers is even more recent. Even the oldest artifacts—which are also the most ambiguous regarding their possible use for numbers—are roughly the same age as what Chomsky estimates for language. There are also contemporary human societies with relatively few numbers. These are mostly concentrated in Australia and South America, a geographic distribution evoking the timeline these continents were peopled, Australia about 60 thousand years ago, the Americas a mere 20 thousand. Both of these sources of evidence suggest that while the perceptual experience of quantity precedes language, language precedes the expression of Bolhuis et al., ‘How Could Language Have Evolved?’. Corballis, ‘The Gestural Origins of Language: Human Language May Have Evolved from Manual Gestures, Which Survive Today as a “Behavioral Fossil” Coupled to Speech’; Hewes, ‘Primate Communication and the Gestural Origin of Language’; Tomasello, Origins of Human Communication. 8 Davidson and Noble, ‘Tools and Language in Human Evolution’; Dibble, ‘The Implications of Stone Tool Types for the Presence of Language during the Lower and Middle Paleolithic’; Higuchi et al., ‘Shared Neural Correlates for Language and Tool Use in Broca’s Area’. 9 Davidson and Noble, ‘The Archaeology of Perception: Traces of Depiction and Language’. 10 Falk, ‘Hominid Paleoneurology’; Holloway, ‘Human Paleontological Evidence Relevant to Language Behavior’; Holloway et al., ‘Evolution of the Brain, in Humans—Paleoneurology’; Tobias, ‘The Emergence of Man in Africa and Beyond’. 6 7

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number. However, there are some important implications. First, even the availability of both the number sense and language does not guarantee that numbers will develop. There are conditions under which numbers do and don’t emerge: Small social groups living in relative isolation from other humans may not develop numbers, while larger groups in more frequent contact with other groups are likely to. It also suggests a dissociation between language and numbers, one consistent with their dissociability in the brain, discussed below, and in writing, discussed in later chapters. The insight that numbers and language are dissociable in terms of brain function and form comes from lesion studies, where damage to particular brain regions caused by strokes or injuries are associated with impairments to specific functions, helping to solidify correlations between function and form. Damage to regions associated with language produce impairments in producing and comprehending speech without concomitant impairment in numeracy, the ability to reason with numbers; conversely, damage to regions associated with numeracy produce impairments in understanding and using numbers without concomitant impairment to language.11 This reversed functional impairment and sparing is called double dissociation, long considered the gold standard in demonstrating independence of brain form and function. Human infants only months old also demonstrate the ability to appreciate quantity,12 suggesting the number sense is innate—not just shared with other species, but determined by factors present from birth, genetically inherited, and fundamental to representational structure.13 The protocol typically used is surprise-them-and-they-look-longer. Infants are shown objects, which are then occluded before reappearing in the same or different quantities. When more or fewer objects appear, infants tend to look longer than they do when objects reappear in the same quantities. Looking longer is interpreted as surprise or interest, the salient point being infants distinguish the quantities in question. The age at which infants can demonstrate quantity appreciation also suggests the number sense is inherently alinguistic, not only because it is shared with so many alinguistic species but because it is found in our own species before infants can speak or have had much exposure to language. The neurological and neurofunctional bases for the double dissociation between numbers and language is well established. Numeracy, numbers, and numerosity are associated with parietal activity,14 finger gnosia,15 and motor-movement planning,16 11 Brannon, ‘The Independence of Language and Mathematical Reasoning’; Varley et al., ‘Agrammatic but Numerate’. 12 Brannon and Roitman, ‘Nonverbal Representations of Time and Number in Animals and Human Infants’; Izard et al., ‘Newborn Infants Perceive Abstract Numbers’; Starkey, Spelke, and Gelman, ‘Numerical Abstraction by Human Infants’; Xu, Spelke, and Goddard, ‘Number Sense in Human Infants’. 13 Carey, The Origin of Concepts. 14 Amalric and Dehaene, ‘Origins of the Brain Networks for Advanced Mathematics in Expert Mathematicians’; Fias et al., ‘Processing of Abstract Ordinal Knowledge in the Horizontal Segment of the Intraparietal Sulcus’; Orban et al., ‘Mapping the Parietal Cortex of Human and Non-Human Primates’.

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while language preferentially involves regions in the frontal and temporal lobes.17 Overlaps, or regions involved in both language and tool-use or numbers and motor movements, complicate the picture. Language and numbers link to different higher-order cognitive domains, and most words for numbers are generated by means of lexical rules, rather than being stored in the mental lexicon, a long-term memory function for the meanings, pronunciations, and syntactic traits of about 10,000 words.18 In fact, number words differ from non-numerical words in quite a few ways. These differences will be discussed in detail in conjunction with the analyses of numerical language and writing in the Ancient Near East. This dissociability, however, has not stopped numbers from being attributed to language. Again, this goes back to the computational model envisioned by Chomsky, who has argued that language can be distinguished from forms of animal communication by its discrete infinity, the ability to generate a near-infinite number of novel combinations from a finite set of sounds and meanings.19 Essentially, numbers represent a subset of this computational capacity: The ‘human number faculty [is] essentially an “abstraction” from human language, preserving the mechanism of discrete infinity and eliminating the other special features of language’.20 Or, it’s possible that both language and numbers are informed by an underlying computational capacity responsible for properties like generativity, with other features added subsequently.21 Supporting this view is the fact that at higher-order levels of syntactic representation, numbers and language share fundamental similarities: Both use fixed sets of rules that govern whether statements are well formed or that transfer properties like truth between statements. In its strong form, the claim has been stated thus: ‘Without language, no numeracy’.22 This does more than position language as necessary but not sufficient because it implies the failure to generate numbers is a cognitive deficiency. In speaking of this failure, Chomsky noted that ‘it is just extraordinarily unlikely that a biological capacity that is highly useful and very valuable for the perpetuation of the species and so on, a capacity that has obvious selection value, should be latent and not used. That would be

15 Penner-Wilger et al., ‘The Foundations of Numeracy: Subitizing, Finger Gnosia, and Fine Motor Ability’; Reeve and Humberstone, ‘Five- to 7-Year-Olds’ Finger Gnosia and Calculation Abilities’. 16 Brooks et al., ‘Abacus: Gesture in the Mind, Not the Hands’; Frank and Barner, ‘Representing Exact Number Visually Using Mental Abacus’. 17 Tremblay and Dick, ‘Broca and Wernicke Are Dead, or Moving Past the Classic Model of Language Neurobiology’. 18 Carreiras et al., ‘Numbers Are Not like Words: Different Pathways for Literacy and Numeracy’. 19 Bolhuis et al., ‘How Could Language Have Evolved?’; Hauser, Chomsky, and Fitch, ‘The Faculty of Language: What Is It, Who Has It, and How Did It Evolve?’. 20 Chomsky, Language and Problems of Knowledge: The Managua Lectures, p. 169. 21 Chomsky, The Generative Enterprise Revisited: Discussions with Riny Huybregts, Henk van Riemsdijk, Naoki Fukui and Mihoko Zushi. 22 Hurford, Language and Number: The Emergence of a Cognitive System, p. 305.

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amazing if it were true’.23 Regarding the lack of numbers in Australian aboriginal languages, linguist James Hurford noted that while ‘all humans appear to have the capacity to acquire a numeral system, only some humans have the attributes or the opportunities which give rise to the development of a numeral system de novo. … Some may be richly endowed with the relevant inventive capacity; others possibly not at all’. 24 Interpretations of these statements range from the trite—obviously, when peoples don’t have numbers, whatever causes numbers to develop didn’t do so—to the racist, positioning peoples without numbers as ‘biologically less than truly human’.25 But the problem is not likely racism as much as it is attributing numbers to language, which assumes they emerge, fully formed, from the computational capacity language is thought to be. This view excludes seeing numbers as concepts that emerge from perceiving and manipulating quantity in material form, with concepts also receiving linguistic form at some point. This alternative perspective pins the development of numbers to the enabling conditions and behaviors being present, rather than computational capacities being fully capable or not. It also allows numbers to change, and thus differ within a particular number system over time and between systems at a time. Language cannot discern or differentiate such conceptual variance because words for numbers like one are assumed to mean the same thing. But even within the history of Western numbers, one has not always been a number: To the ancient Greeks, one was the unity, a metaphysical construct that differed from all the other numbers, which represented plurality. One did not become a number just like all the others until only within the last several centuries. As recently as 1728, encyclopedist Ephraim Chambers would observe: ’Tis difputed among Mathematicians, whether or no Unity be a Number.—The generality of Authors hold the Negative ; and make Unity to be only inceptive of Number, or the Principle thereof ; as a Point is of Magnitude, and Unifon of Concord. Stevinus [mathematician Simon Stevin, who helped influence one to become a number like the others in the 16th century] is very angry with the Maintainers of the Opinion: and yet, if Number be defin’d a Multitude of Unites join’d together, as many Authors define it, ’tis evident Unity is not a Number’.26

DEVELOPMENT AND NUMBERS From language we go to developmental acquisition, which brings us to Piaget, a scholar whose work has been particularly influential in numerical cognition, especially for the Ancient Near East. Jean Piaget was a clinical psychologist best known for his work

Chomsky, The Generative Enterprise Revisited, pp. 18–19. Hurford, Language and Number, p. 73. 25 Verran, ‘Aboriginal Australian Mathematics: Disparate Mathematics of Land Ownership,’ p. 292. 26 Chambers, Cyclopædia, or, An Universal Dictionary of Arts and Sciences, Vol. 2, p. 323. 23 24

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on cognitive development in children.27 Noting differences between childish thinking and adult thought, Piaget envisioned a four-stage progression from one to the other.28 Children from birth to the age of 2 were in the sensorimotor stage, experiencing the world through sensation and movement. Years 2 to 7 were the preoperational stage, in which childish thinking was characterized by an absence of both concrete logic and the ability to manipulate information mentally. Years 7 to 12 were the concrete operational stage, where children became able to think logically but only concretely so, in being restricted to matters involving the physical manipulation of objects. Finally, children 12 and older reached the formal operational stage, able to think abstractly, solve complex problems, and be aware of and understand their own thoughts. These stages were necessarily progressive because the earlier ones had to be in place before any of the later ones could develop. Regarding numbers, Piaget envisioned acquisition as a process of developing biologically predetermined cognitive structures through the experience of objects. However, the cognitive structures that would ultimately develop in numbers would have the same final form regardless of the content of the experience that prompted them.29 Beyond mentioning the representationalism this implies, further elaboration of Piagetian developmental theory is tangential to the present argument. More important is mentioning two interrelated matters, the implications of applying developmental theories like Piaget’s to entire societies and how ontogenetic aspects of conceptual acquisition might overlie the historical development of concepts. The former has specifically influenced the view of how numeracy developed in Ancient Near Eastern societies, mainly though the work of psychologist and historian Peter Damerow, something I discuss in later chapters. Piaget drew upon earlier work by sociologist Lucien Lévy-Bruhl30 to extend his ideas about cognitive ontogenesis in children to entire societies. For both these scholars, societies possessed distinct mentalities, just as children and adults did. Piaget divided societies into two groups. The first, which contained traditional and archaic societies, he labeled primitive and as having childish thinking; the second, consisting of modern, industrialized societies like his own, he deemed rational, scientific, civilized, and as possessing adult thought. Such pejorative characterizations were not uncommon in the 19th century; they continued well into the 20th century and, as shown in discussing the idea that numbers originate in language, to uncomfortably recent times as well, albeit in weaker form. Piaget construed many parallels between the so-called primitive mentality of the ‘civilisations inférieures’ and the immature ‘mentalité enfantine’31: 27 A version of some of the material that follows was published in Overmann, ‘Updating the “Abstract–Concrete” Distinction in Ancient Near Eastern Numbers’. 28 Inhelder and Piaget, The Growth of Logical Thinking: From Childhood to Adolescence; Piaget, The Child’s Conception of Number; The Construction of Reality in the Child. 29 Nicolopoulou, ‘The Invention of Writing and the Development of Numerical Concepts in Sumeria: Some Implications for Developmental Psychology’, p. 207. 30 Lévy-Bruhl, L’âme Primitive; Les Fonctions Mentales dans les Sociétés Inférieures. 31 Piaget, ‘Logique Génétique et Sociologie’, pp. 191–201.

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On peut citer comme exemples la tendance à l’affirmation sans preuve, le caractère affectif de la pensée, son caractère global, non analytique (le syncrétisme), l’absence de cohérence logique (des principes de contradiction et d’identité considérés comme des structures formelles), la difficulté à manier le raisonnement déductif et la fréquence des raisonnements par identification immédiate (participation), la causalité mystique, l’indifférenciation du psychique et du physique, la confusion du signe et de la cause, du signe et de la chose signifiée, etc. Nous ne prétendons nullement, cela va sans dire, que chacun de ces traits se présente de la même manière chez le primitif et l’enfant, et il faudrait un volume pour marquer les nuances, pour souligner l’aspect fonctionnel des ressemblances et écarter les identifications brutales. Mais, dans les grandes lignes, nous pensons qu’il y a des analogies.32 [Examples include the tendency to assert without proof, the affective character of thought, its global, non-analytical character (syncretism), the absence of logical coherence (principles of contradiction and identity considered as formal structures), the difficulty in handling deductive reasoning and the frequency of reasoning by immediate identification (participation), mystical causality, the undifferentiation of the psychic and of the physical, the confusion of the sign and the cause, the sign and the thing signified, etc. We do not claim, it goes without saying, that each of these features is presented in the same way in the primitive and the child, and it would require a volume to mark the nuances, to underline the functional aspect of the similarities and to exclude the brutal identifications. But in general, we think there are analogies.]

Particularly important to the present argument is the idea that the sign is confused with the thing signified, highlighted in bold. Piaget did identify the two as having separate mechanisms, with ‘tradition sociale’ constraining the primitive mentality and ‘l’égocentrisme de la pensée’ that of the child; together, these were ‘l’obstacle principal à la mise en parallèle du primitif et de l’enfant’.33 Piaget was describing real phenomena, using the language of his time. Certainly, there are meaningful cognitive differences between societies, though hardly ones warranting the labels used in 19th-century discourse. Contemporary psychology is just now beginning to recognize cognitive differences between individuals from societies that are Western, educated, industrialized, democratic, and rich—WEIRD—and societies that are not WEIRD. This distinction is an important one because WEIRD people are only a small portion of the world’s population, but they skew our understanding of what human cognition is like because they perform most of the experiments published in the psychological literature and use WEIRD college students as test subjects. Some of the WEIRD–non-WEIRD differences are surprising: One is shown with the Müller-Lyer illusion, where lines of identical length are perceived as longer or shorter depending on whether their end arrows point out or in. People from WEIRD societies are reportedly more prone to seeing lines with unequal length, an effect on visual perception thought 32 33

Piaget, ‘Logique Génétique et Sociologie’, p. 194; emphasis added. Piaget, p. 192.

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related to habituation to urban linearity.34 That is, cognitive differences reflect enculturation into material differences—‘the human brain is as much a cultural artifact as a biological entity’.35 For numbers, brains enculturated into different cultural systems appear to process identical numerals differently, a phenomenon language cannot fully explain.36 However, the perception of non-symbolic quantity appears impervious to the WEIRD–non-WEIRD divide.37 However terminologically gauche, Piaget was addressing the question of how societies derive truth from opinion. He viewed this as a development in societal thinking that involved, as ontogenetic maturation does, acquiring concepts and constructing structures of thought. There are plausible similarities between the two processes. Both involve interacting with social others and the physical world. Both are progressive in the sense that later constructions depend on previous ones. And both produce cognitive outcomes that are relatively consistent when viewed across individuals and societies. Some similarity is only to be expected, though not necessarily because of innate representational structures in the mind: A society is composed of individuals enculturated from birth to reproduce and transmit its behaviors, knowledge, and manner of thinking. Enculturation simply incorporates new individuals as seamless parts of the same cultural fabric. While Piaget differentiated individual and social developmental mechanisms, he does not appear to have made a similar distinction between the child’s acquisition of existing social knowledge, which entails the preexistence of concepts, terminology, and knowledgeable others, from its social invention. I would argue that when new knowledge is generated, adult activity likely dominates discovery, transmission, and conventionalization. Certainly, since the developmental acquisition of number involves ontogenetic maturation, children are unlikely to be a significant factor in numerical elaboration—they just don’t yet have the cognitive wherewithal to understand the concepts, let alone invent new ones. However, the requirement to explain concepts in terms they can understand may plausibly relate, at least in some small part, to numerical explication, which is part of elaboration. The processes of acquisition and invention undoubtedly overlap, as the latter involves knowledge and skills acquired when young that are perhaps organized ontogenetically in ways that facilitate creativity, analogous to the way children turn pidgin into creole in language. And both adults and children participate in the transmission and conventionalization of new knowledge. For numbers, a vast literature covers how maturational change affects numerical acquisition and numeracy. This literature is much less concerned, if at all, with diachronic change in the number systems themselves. I do not spend any more time discussing developmental aspects of numbers here, beyond acknowledging that children acquire knowledge of, behaviors associated with, and ways of thinking about numbers both informally, through language and everyday use, and formally, through education. Henrich, Heine, and Norenzayan, ‘The Weirdest People in the World?’, pp. 64–65. Malafouris, How Things Shape the Mind: A Theory of Material Engagement, p. 45. 36 Tang et al., ‘Arithmetic Processing in the Brain Shaped by Cultures’. 37 Henrich, Heine, and Norenzayan, ‘The Weirdest People in the World?’, p. 69. 34 35

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The similarities and overlaps between the child’s acquisition of existing knowledge and a society’s generation of new knowledge may well have further relevance to the history of thought.38 It is orthogonal, however, to discussing how numbers are realized and elaborated, for the reasons I have already mentioned. Before I shift gears to abstraction and categorization as cognitive processes critical to realizing and elaborating numbers, I offer this final point: Piaget is hardly the last word in ontogenetic change in numbers. Besides investigating how acquisition works for most children, the developmental literature also looks at proficiency, dysfunction, and the role of language. How features of language assist typical acquisition is an area where the focus on WEIRD societies to the exclusion of those that are not WEIRD resurfaces. An example is grammatical number, distinctions of singulars and plurals demonstrated to help children acquire concepts of number. Not all languages have grammatical number, however, and there are highly numerate societies with languages that lack this feature, like Cantonese. This suggests that while grammatical number is useful to acquisition when present, its absence is no barrier to acquisition or elaboration. I discuss grammatical number in more detail in Chapter 6, for it provides an interesting window into ancient numeracy.

CATEGORIZATION AND ABSTRACTION Categorization is the cognitive process that groups or differentiates objects according to the similarities or dissimilarities of their properties, relations, or functions (see Table 4.1). Non-human species can categorize to some degree, but humans also differ significantly in this ability.39 Many species, including humans, make categorical judgments of identity, discerning sameness or difference between single elements. Unlike most species, humans also make categorical judgments of relations, sameness and difference in multiple elements. Non-human apes demonstrate the ability to judge relations, but they have to be specifically trained to do it; even with training, they don’t categorize relations spontaneously or achieve the same levels of hierarchical complexity as humans do.40 In essence, the more properties objects share, the easier categorization becomes; conversely, the fewer shared properties, the more difficult. It is doubtful chimpanzees could learn to distinguish cross-dimensional relations, which require suppressing nonessential, highly salient properties to focus on pertinent, less salient ones—like quantity.

38 Oesterdiekhoff, ‘Is a Forgotten Subject Central to the Future Development of Sciences? Jean Piaget on the Interrelationship between Ontogeny and History’. 39 Christie and Gentner, ‘Relational Similarity in Identity Relation: The Role of Language’; Gentner and Colhoun, ‘Analogical Processes in Human Thinking and Learning’. 40 Langer, The Origins of Logic: One to Two Years; Thompson and Oden, ‘Categorical Perception and Conceptual Judgments by Nonhuman Primates: The Paleological Monkey and the Analogical Ape’; Thompson, Oden, and Boysen, ‘Language-Naive Chimpanzees (Pan troglodytes) Judge Relations between Relations in a Conceptual Matching-to-Sample Task’.

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Definition Sameness or difference between single elements Sameness and difference in multiple elements Sameness and difference in only some properties of multiple elements

Example A is the same as A, and A is not the same as B.

Found In Humans; many other species

AA and BB both consist of two identical elements.

Humans; other great apes, to a limited extent Only humans (that we know of)

aaaaaaaa and BBB both consist of identical elements; aaaaaaaa and cde share small letter size; BBB and cde are both trios.

Table 4.1. Categorical judgments. The differences across species suggest only humans would form concepts of numbers. Data compiled and adapted from Christie and Gentner, ‘Relational Similarity in Identity Relation: The Role of Language’; Gentner and Colhoun, ‘Analogical Processes in Human Thinking and Learning’.

Why does our ability to categorization exceed that of even closely related species? One reason is our brains are simply larger, and brain size, when adjusted for body size, correlates with intelligence.41 Another reason is our ability to exploit materiality for cognitive purposes. For concepts, material anchoring and stabilizing give concepts a continuity beyond what brains can achieve on their own.42 Ignoring obvious properties while attending to less salient ones means attending not just to wholes but to parts, evoking ancestral concerns with parts of tools, something discerned in the archaeological record as change in the design features of stone tools.43 Our complex hierarchical categorization may simply be a consequence of our ability to subdivide material forms. Writing, another material technology, lets us gather, co-locate, manipulate, and analyze large bodies of knowledge for their similarities and differences, implying a mechanism for categorical sharpening.44 Considered cross-culturally, categorizing is similarly complex across and despite significant differences of material culture. What does vary cross-culturally are the kinds of objects and entities categorized; the organizing principles used for classification; whether categories are based on properties, relations, or functions; and the amount and boundaries of the categories formed.45 This suggests the differences between human and non-human species in the ability to categorize lie in material engagement per se, not Falk, ‘Evolution of the Primate Brain’. Fauconnier and Turner, ‘Conceptual Integration Networks’; Hutchins, ‘Material Anchors for Conceptual Blends’. 43 Gowlett, ‘The Elements of Design Form in Acheulian Bifaces: Modes, Modalities, Rules and Language’; Overmann and Wynn, ‘Materiality and Human Cognition’. 44 Veldhuis, History of the Cuneiform Lexical Tradition; Watson and Horowitz, Writing Science before the Greeks: A Naturalistic Analysis of the Babylonian Astronomical Treatise MUL.APIN. 45 Hunn and French, ‘Alternatives to Taxonomic Hierarchy: The Sahaptin Case’; Sillitoe, ‘Contested Knowledge, Contingent Classification: Animals in the Highlands of Papua New Guinea’; Unsworth, Sears, and Pexman, ‘Cultural Influences on Categorization Processes’. 41 42

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in specific forms of materiality. The independence of technological complexity and categorization, in turn, suggests expansion of the ability may have begun early in the hominid lineage—the category including all species of bipedal apes, whether they are extinct or extant—perhaps when they first began to use tools several million years ago. Russell’s logical types suggests how categorization might work in numbers, and why other species are unlikely to have concepts of numbers in the same way we do, even though they share with humans the ability to appreciate quantity in small sets of objects. When individual objects (Type 1) are compared, they are categorized as similar or dissimilar through judgments of identity, something non-human species can do. Judgments of identity allow similar objects to be grouped into sets of like objects (Type 2), something non-human primates can do that reflects their ability to manipulate material forms in addition to their capacity for categorizing. Comparing cardinality across sets of objects (Type 3) requires the ability to judge categorical relations, something non-human primates can’t do, perhaps because it requires suppressing salient, nonrelevant qualities to focus on unobvious but relevant ones. The judgment that a pair of cars and two fingers share two-ness—the number two—requires suppressing all the overt properties that differentiate cars from fingers to discern what is not nearly as obvious, their shared quantity. This likely makes judgments of shared cardinality something only humans achieve. The categories formed by classifying objects by their properties are abstractions, and abstraction is both process and product. As process and product, abstraction forms concepts by generalizing, decontextualizing, synthesizing, and reifying.46 Generalization identifies properties common to sets of objects, inducts from particulars, and applies inductive insights to new domains. Decontextualization extracts content from its original circumstances to remove their influence on its meaning. Synthesis combines parts to form wholes, often in such a way that sums are greater than their constitutive parts. Reification reinterprets processes or relations as permanent entities in their own right, making them available to act as inputs to other processes or relations. Reification is thought to be particularly challenging, since it requires distinguishing and then separating products from their instantiating processes.47 As a cognitive process and its products, abstraction tends to be thought of as a purely mental activity. This representationalist view tends to exclude the recognition that both process and product are at least as much behavioral and material as they are mental. Realizing concepts of number from shared cardinalities and extending them to new sets of objects imply behaviors with material forms. Similarly, dissociating concepts from the objects and sets giving rise to them depend on sequencing and comparing objects for purposes like counting, performing counting away from the enumerated objects themselves by means of counters, or leveraging material properties as proxies 46 Dreyfus, ‘Advanced Mathematical Thinking Processes’; Ferrari, ‘Abstraction in Mathematics’; Sfard and Linchevski, ‘The Gains and the Pitfalls of Reification? The Case of Algebra’. 47 Gray and Tall, ‘Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic’.

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for new conceptual relations. Concepts like number, counting, and the count, and opportunities to notice and explicate patterns and relations, can emerge from material engagements like making notches or rearranging beads. Essentially, behaviors with material forms provide opportunities for our brains to do what they do best, recognize patterns and forget details that are less important. Abstract is one of those terms whose meaning depends on its context. For psychologists, it means a cognitive process and its conceptual products. For mathematicians, it has the sense of objects that are neither mental nor material and that lack spatiotemporal locus, causal efficacy, or both.48 The term can be used in other ways as well: It can connote something that has become distilled, refined, or purified to its essential nature, in the process perhaps becoming truer or more accurate. It can mean something has achieved a state more rarified and complex than the one preceding it, becoming more difficult for the uninitiated to comprehend. Abstract differentiates the theoretical or pure mathematics interested in what numbers are from their everyday, mundane use to answer questions like how many and how much; this sense has been used to elevate Greek mathematics above its earlier Mesopotamian counterpart.49 Abstract can mean specified or unspecified, or connected or not to the objects of which a number is the quantity, a meaning with particular significance in Ancient Near Eastern numeracy.50 It can be used to characterize how little a picture resembles the object it depicts, the sense in which Schmandt-Besserat first used it in discussing change in early Mesopotamian writing.51 I use the term in still yet another way to discuss what happens when numbers are used with multiple material devices: As concepts, they become functionally independent of all the material forms used to represent and manipulate them.52 These usages are detailed further in later chapters.

FINGER-COUNTING AND HUMAN NEUROANATOMY Finger-counting leverages specific neuroanatomical structures and functions. One is the interaction between the region that subserves the number sense and the parts involved in using tools, ‘knowing’ the fingers, recalling arithmetic facts, and expressing bodily experience in language. Another is the sheer amount of cortex dedicated to the sensation and movement of the human hand, and the way it is topographically organized to influence finger-counting—and hence, numbers—toward linearity. A final item is the involvement of motor-planning functions for hand movements in so-called mental arithmetic. Cumulatively, these demonstrate the strong somatic basis for expressing numbers by means of the fingers. Quite simply, we count on our fingers because we are wired to do so, and this is independent of how elaborated our number system might be. And besides influencing linearity, using the digits of our pentadactyl Linnebo, ‘Platonism in the Philosophy of Mathematics’; Rosen, ‘Abstract Objects’. Høyrup, ‘Sub-Scientific Mathematics: Observations on a Pre-Modern Phenomenon’. 50 Robson, ‘Mesopotamian Mathematics’, p. 75. 51 Schmandt-Besserat, ‘The Earliest Precursor of Writing’, p. 50. 52 Overmann, ‘Updating the “Abstract–Concrete” Distinction in Ancient Near Eastern Numbers’, pp. 9–10. 48 49

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limbs as a material structure for representing numbers patterns how we group them— very often, by tens, fives, and twenties. The neuroanatomic basis for finger-counting starts with the part of the brain that perceives quantity, neuroanatomy consistent across primate species.53 From monkeys to great apes, the category that includes humans, the number sense is a function of the intraparietal sulcus, a shallow cortical groove located toward the top of the parietal lobe, one of the four major parts of the cerebrum. In humans, the parietal lobe has been implicated in an astonishing variety of functions that include, in addition to numerosity and numeracy, detailed vision and tool-use, ‘knowing’ the fingers, motor planning, perceptual–motor coordination, spatiality, multimodal association, the sense of self and self-representation, various forms of attention, working memory, aspects of long-term memory and language, and interpreting intent.54 Beyond its involvement in these multiple functions, the human parietal lobe is remarkable in other ways. Its evolutionary expansion in humans is perhaps the single characteristic that best distinguishes our brains from the brains of other primates, including those of the most closely related species, the Neandertals.55 While hominid brains became bigger overall in relation to body size, in our lineage, parietal lobes be53 Biro and Matsuzawa, ‘Use of Numerical Symbols by the Chimpanzee (Pan troglodytes)’; Choi et al., ‘Cytoarchitectonic Identification and Probabilistic Mapping of Two Distinct Areas within the Anterior Ventral Bank of the Human Intraparietal Sulcus’; Le Gros Clark, Cooper, and Zuckerman, ‘The Endocranial Cast of the Chimpanzee’; Lewis and Van Essen, ‘Corticocortical Connections of Visual, Sensorimotor, and Multimodal Processing Areas in the Parietal Lobe of the Macaque Monkey’; Tomonaga, ‘Relative Numerosity Discrimination by Chimpanzees (Pan troglodytes): Evidence for Approximate Numerical Representations’; Varga, Pavlova, and Nosova, ‘The Counting Function and Its Representation in the Parietal Cortex in Humans and Animals’. 54 Ansari et al., ‘Neural Correlates of Symbolic Number Processing in Children and Adults’; Bruner, ‘Morphological Differences in the Parietal Lobes within the Human Genus’; Bruner et al., ‘Midsagittal Brain Variation and MRI Shape Analysis of the Precuneus in Adult Individuals’; Cantlon et al., ‘Functional Imaging of Numerical Processing in Adults and 4-Y-Old Children’; Diester and Nieder, ‘Complementary Contributions of Prefrontal Neuron Classes in Abstract Numerical Categorization’; Hamilton and Grafton, ‘Goal Representation in Human Anterior Intraparietal Sulcus’; Hubbard et al., ‘Numerical and Spatial Intuitions: A Role for Posterior Parietal Cortex?’; Koenigs et al., ‘Superior Parietal Cortex Is Critical for the Manipulation of Information in Working Memory’; Orban et al., ‘Mapping the Parietal Cortex of Human and Non-Human Primates’. 55 Bruner, ‘Geometric Morphometrics and Paleoneurology: Brain Shape Evolution in the Genus Homo’; ‘Morphological Differences in the Parietal Lobes within the Human Genus’; Bruner and Holloway, ‘A Bivariate Approach to the Widening of the Frontal Lobes in the Genus Homo’; Bruner, de la Cuétara, and Holloway, ‘A Bivariate Approach to the Variation of the Parietal Curvature in the Genus Homo’; Bruner, Manzi, and Arsuaga, ‘Encephalization and Allometric Trajectories in the Genus Homo: Evidence from the Neandertal and Modern Lineages’; Bruner et al., ‘Midsagittal Brain Variation and MRI Shape Analysis of the Precuneus in Adult Individuals’.

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came even bigger still in relation to the other lobes. This parietal expansion in H. sapiens was also vertically oriented, causing the overall shape of the human brain to become more globular, or rounder than is typical for primate brains. Again, this includes the Neandertals, whose larger brains were more prototypically elongated. And because there is a general relation between the size of a brain region and the adaptive use of its functions, parietal expansion in our lineage can be linked to important aspects of our ecological niche, like tool-use,56 a behavioral difference between H. sapiens and the Neandertals.57 Parietal expansion had important consequences for our species cognition. First, the rounder shape was advantageous. Long pathways between brain regions are expensive, metabolically speaking, so these connections tend to shorten as brains become larger—but, globularization effectively decreased the length they needed to be.58 This can be envisioned by contrasting the internal length connecting the poles of an oblong shape like a football with that needed to connect any two opposing points on a sphere with the same volume: The former involves longer connections than the latter. Within regions, the rounder shape would have condensed neural pathways, increasing withinregion connectivity.59 Shorter, denser pathways are faster and more efficient,60 an advantage for any brain, but especially in mitigating the evolutionary trade-offs associated with larger ones. Parietal expansion would also have co-located regions associated with numbers, like the intraparietal sulcus, with those associated with the fingers, tool-use, and language functions like metaphor and inner speech,61 facilitating their interaction. Human numeracy thus appears to be, at least in part, a consequence of evolutionary morphological change in human brains related to tool-use.62 The intraparietal sulcus interacts with the supramarginal and angular gyri, adjacent ridges on the surface of the parietal lobe that are active when mathematical tasks are

56 Orban and Caruana, ‘The Neural Basis of Human Tool Use’; Orban et al., ‘Mapping the Parietal Cortex of Human and Non-Human Primates’; Rehkämper, Frahm, and Mann, ‘Brain Composition and Ecological Niches in the Wild or under Man-Made Conditions (Domestication): Constraints of the Evolutionary Plasticity of the Brain’. 57 Wynn, Overmann, and Coolidge, ‘The False Dichotomy: A Refutation of the Neandertal Indistinguishability Claim’. 58 Kaas, ‘Why Is Brain Size so Important: Design Problems and Solutions as Neocortex Gets Bigger or Smaller’, p. 17. 59 Rilling and Insel, ‘Differential Expansion of Neural Projection Systems in Primate Brain Evolution’. 60 Gibson, ‘Myelination and Behavioral Development: A Comparative Perspective on Questions of Neoteny, Altriciality and Intelligence’; Gibson and Petersen, ‘Introduction’. 61 Coolidge and Overmann, ‘Numerosity, Abstraction, and the Emergence of Symbolic Thinking’; Geva et al., ‘The Neural Correlates of Inner Speech Defined by Voxel-Based Lesion Symptom Mapping’; Seghier, ‘The Angular Gyrus: Multiple Functions and Multiple Subdivisions’. 62 Coolidge and Wynn, The Rise of Homo sapiens: The Evolution of Modern Thinking.

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performed.63 The supramarginal gyrus has been implicated in tool-use,64 as well as inner speech, the internal experience of conscious thought as coherent language.65 The angular gyrus has been implicated in multiple functions contributing to numeracy. These include metaphorizing, expressing bodily experience and numerical concepts in language,66 manipulating numbers in verbal form, and retrieving arithmetic facts from memory.67 Linking the angular gyrus with both numbers and the fingers are finger gnosia and finger-counting.68 Finger gnosia, the ability to ‘know’ the fingers, predicts mathematical ability: A person who can tell which finger has been tapped when she cannot see her hands is more likely to be good at mathematical tasks.69 Damage to the angular gyrus is also associated with finger agnosia, impaired ability to ‘know’ the fingers; acalculia, impaired ability to perform even simple mathematical tasks; and impaired ability to use the fingers for counting.70

63 Dehaene et al., ‘Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence’; Zamarian, Ischebeck, and Delazer, ‘Neuroscience of Learning Arithmetic: Evidence from Brain Imaging Studies’. 64 Orban and Caruana, ‘The Neural Basis of Human Tool Use’. 65 Ardila, ‘There Are Two Different Language Systems in the Brain’; Geva et al., ‘The Neural Correlates of Inner Speech Defined by Voxel-Based Lesion Symptom Mapping’; Lurito et al., ‘Comparison of Rhyming and Word Generation with FMRI’; McGuire et al., ‘Functional Anatomy of Inner Speech and Auditory Verbal Imagery’; Owen, Borowsky, and Sarty, ‘FMRI of Two Measures of Phonological Processing in Visual Word Recognition: Ecological Validity Matters’. 66 Lakoff and Núñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. 67 Dehaene et al., ‘Three Parietal Circuits for Number Processing’; Grabner et al., ‘Individual Differences in Mathematical Competence Predict Parietal Brain Activation during Mental Calculation’; ‘To Retrieve or to Calculate? Left Angular Gyrus Mediates the Retrieval of Arithmetic Facts during Problem Solving’; ‘Fact Learning in Complex Arithmetic and FiguralSpatial Tasks: The Role of the Angular Gyrus and Its Relation to Mathematical Competence’; Ramachandran, A Brief Tour of Human Consciousness: From Impostor Poodles to Purple Numbers. 68 Penner-Wilger et al., ‘The Foundations of Numeracy’; Reeve and Humberstone, ‘Fiveto 7-Year-Olds’ Finger Gnosia and Calculation Abilities’; Roux et al., ‘Writing, Calculating, and Finger Recognition in the Region of the Angular Gyrus: A Cortical Stimulation Study of Gerstmann Syndrome’. 69 Gracia-Bafalluy and Noël, ‘Does Finger Training Increase Young Children’s Numerical Performance?’; Marinthe, Fayol, and Barrouillet, ‘Gnosies Digitales et Développement des Performances Arithmétiques’. 70 Roux et al., ‘Writing, Calculating, and Finger Recognition in the Region of the Angular Gyrus’.

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Fig. 4.2. The cortical homunculus. Penfield’s original drawing characterizes both the proportion of primary cortex dedicated to motor functions and its topographical organization. Penfield’s idea of how the cortices were organized was based on the movements produced when the cortices of anesthetized animals were electrically stimulated;71 it has since been confirmed by magnetic stimulation studies with conscious human subjects, as well as lesion studies correlating motor cortex injuries with impaired muscle control. Adapted from Penfield and Rasmussen, The Cerebral Cortex of Man: A Clinical Study of Localization of Function, p. 215, Fig. 115, and an image in the public domain.

The hand is more than ‘simply an instrument for manipulating an externally given objective world by carrying out the orders issued to it by the brain; it is instead one of the main perturbatory channels through which the world touches us, and it has a great deal to do with how this world is perceived and classified’.72 The hand, in fact, is both instrument and actor, capable of both sensing and manipulating the world in ways that few species share, apart from other primates; this quality connects our internal and external domains of experience.73 As you might expect from this bipolarity, much of the human brain is dedicated to the hands and fingers: knowing where they are, what they are feeling and doing, and controlling their movements. The proportion of cortex ded71 Penfield and Jasper, Epilepsy and the Functional Anatomy of the Human Brain; Penfield and Rasmussen, The Cerebral Cortex of Man. 72 Malafouris, How Things Shape the Mind, p. 60. 73 Gallagher, ‘The Enactive Hand’.

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icated to their sensory activity and motor control is nicely illustrated by the Penfield homunculus, a tiny human figure whose hands are larger than his head and body combined. Penfield’s original drawings (Fig. 4.2) additionally characterized the human sensorimotor cortices in terms of their topographical layout (Fig. 4.3), a function of how bodies grow from fertilized ovum, wherein cells are added in a head-to-tail sequence organized and governed by the DNA ‘blueprint’. Topographical organization is advantageous in allowing our motor and sensory domains to be represented ‘continuously and completely’ in a coherent map, rather than one disconnected and discontinuous across anatomical boundaries.74

Fig. 4.3. Topographical layout of the motor and somatosensory cortices. The insert (far left) shows the respective location of the motor and sensory cortices. The sensory diagram (right) is broadly similar but not identical to the motor diagram (left), primarily because it includes body parts like the nose and teeth that have sensation but little movement. Note the linear arrangement of the mapping for the fingers in this diagram and Fig. 4.2. Adapted from images in the public domain.

Finger-counting and its expansion as body-counting, a kind of tally counting found today in Papua New Guinea, leverage the topographical organization of our sensorimotor cortices, extending the brain and body’s inherently linear organization to numbers as they are counted on the fingers and body. This organization is not destiny for how we count on the fingers, however, since even within societies that count from one side of the hand to the other, there is significant variability in matters like whether counting is initiated with the right or left hand, whether the thumb or little finger is used to start or end the sequence, whether the thumb is considered part of the sequence, whether the fingers are bent, straightened, or tapped to indicate their use, and 74 Patel, Michael, and Snyder, ‘Topographic Organization in the Brain: Searching for General Principles’, p. 351.

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whether the fingers, finger segments, or spaces between the fingers are used. This variability likely occurs because of the hand’s bipolarity: It’s experienced from both the inside and the out. What unites all such variants is our cross-cultural tendency to use the fingers for counting, select recurrent points for starting and stopping, and proceed between the two in recurrent fashion. This tendency not only capitalizes on our physiology, it also minimizes the demands on cognitive resources like working memory and attention while improving the accessibility and reliability of the numerical information we instantiate with our fingers. Simply, we use the hand as an instrument for counting partly because of our internal physiological ‘wiring’, and partly because we visually perceive the fingers and adapt how we use the hand for counting like we would any other material device. The number sense itself is topographically structured as well: Lateral and medial regions of the posterior parietal lobe respond preferentially to high and low nonsymbolic quantities, respectively.75 The region of the brain demonstrating this structure is broadly similar to, but not identical with, the intraparietal sulcus. While sensory organs in general are topographically organized as a function of how bodies grow from single cells, no sense organ is associated with the number sense, which in fact spans at least three—vision, hearing, and touch. The reason why the number sense is topographically organized is currently unclear, but its benefits are not. The ‘map’ created by the brain’s topographic layout of quantity appreciation may interact with visuospatially appreciated quantity stimuli, perhaps underlying the so-called mental number line76 and predisposing us to order quantities by their magnitude. Its functionality may overlap with the topographical mapping of object size, ‘potentially allowing consideration of both quantities together when making decisions’77 and supporting ‘the representation of higher-order abstract features in the association cortex’.78 The planning of motor movements, but not necessarily their being carried out, appears essential to so-called mental computation. Evidence for this view comes from three sources. The first is mental abacus, a technique for performing arithmetical operations by manipulating an imaginary device. Practitioners of the technique move their fingers, hands, and arms to perform calculations just as if they slid actual beads along the rods of a physical abacus. I have seen children demonstrate the technique to a roomful of awed observers, adding and subtracting long strings of multi-digit numbers, calculating the square roots of six-digit numbers, and multiplying 10-digit numbers. Mental abacus discards language in favor of movement and memory, especially visual working memory.79 Interestingly, physical movements do not need to be seen visually 75 Harvey et al., ‘Topographic Representation of Numerosity in the Human Parietal Cortex’, p. 1124. 76 Harvey et al., p. 1126. 77 Harvey et al., ‘Topographic Representations of Object Size and Relationships with Numerosity Reveal Generalized Quantity Processing in Human Parietal Cortex’, p. 13525. 78 Harvey et al., ‘Topographic Representation of Numerosity in the Human Parietal Cortex’, p. 1123. 79 Frank and Barner, ‘Representing Exact Number Visually Using Mental Abacus’.

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or carried out physically. Performance is unimpaired when practitioners perform blindfolded or keep their hands still; however, requiring unrelated hand movements dramatically interferes with performance.80 This suggests the critical element is motormovement planning, rather than its execution, both functions of the motor cortex, a region of the frontal lobe. A second source of evidence for the involvement of motor-movement planning in ‘mental’ calculation is gesture, a communicative mode integrated with both language and numbers in some interesting ways. Gesture makes our numerical and spatial intuitions accessible, both to ourselves and to others, by making them visible. It expresses through iconic means ideas we cannot verbalize, either because we lack the explicit concepts, the necessary vocabulary, or both. It adds meaning to what we can express in language, not just emphasizing but illustrating, diagrammatically as well as numerically. Gesture also reduces demands on memory and improves learning: Children who gesture are more likely to solve previously difficult mathematical tasks and incorporate novel problem-solving strategies.81 Physical movement involves both the planning and execution of motor movements, and in the case of gesture, may communicate numerical insights that are not verbal in nature. The third source of evidence is found in the cerebellum, the ‘little brain’ located at the top of the brain stem. Though only about 10 percent of the total brain, the cerebellum contains roughly half its neurons. Traditionally, the cerebellum been understood as having a role in learning, sequencing, and controlling motor movements, especially the fine movements needing greater control. It is now thought to also support our abilities to recognize patterns, form and manipulate abstract concepts, and make higher-order decisions and rules.82 Emerging research shows the cerebellum may also provide a ‘common computational language to movement and cognitive processes (including mathematics)’.83 The idea is that as movements and mental processes are repeated, the cerebellum creates efficient internal models that effectively bypass the more time-

Brooks et al., ‘Abacus’. Broaders et al., ‘Making Children Gesture Brings out Implicit Knowledge and Leads to Learning’; Cook, Mitchell, and Goldin-Meadow, ‘Gesturing Makes Learning Last’; Cook, Yip, and Goldin-Meadow, ‘Gesturing Makes Memories That Last’; Goldin-Meadow, Cook, and Mitchell, ‘Gesturing Gives Children New Ideas about Math’; Goldin-Meadow et al., ‘Explaining Math: Gesturing Lightens the Load’; Ping and Goldin-Meadow, ‘Gesturing Saves Cognitive Resources When Talking about Nonpresent Objects’. 82 Balsters et al., ‘Cerebellum and Cognition: Evidence for the Encoding of Higher Order Rules’; Koziol, Budding, and Chidekel, ‘Adaptation, Expertise, and Giftedness: Towards an Understanding of Cortical, Subcortical, and Cerebellar Network Contributions’; Vandervert, ‘The Appearance of the Child Prodigy 10,000 Years Ago: An Evolutionary and Developmental Explanation’; Vandervert, Schimpf, and Liu, ‘How Working Memory and the Cerebellum Collaborate to Produce Creativity and Innovation’. 83 Vandervert, ‘The Origin of Mathematics and Number Sense in the Cerebellum: With Implications for Finger Counting and Dyscalculia’, p. 4. 80 81

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consuming original cortical circuits to make both movements and mental processes ‘smoother, quicker, and progressively more error-free’.84 The observations that an abacus need not be physically present, nor finger movements seen or carried out during calculation, may reflect the recruitment of motor-planning functions in handling tools over an evolutionary span of time—well over 3 million years in the hominid lineage.85 This has an important implication for mental content: Our engagement of material forms may be the basis for that content, even in the absence of both material forms and the movements needed to engage them. That is, even without materiality and movement, our brains function as if these components were present. These neural muscles86 may explain phenomena like daydreaming and imagination, as well as the participation of mobility-impaired individuals in numeracy. The interesting question for numbers is whether the involvement of motor-planning functions reflect our evolutionary history for using tools generally, or counting technologies specifically. I suspect it to be a bit of both. At this point in building a coherent picture of numerical origins, I have drawn mainly from the field of psychology. I reviewed how the number sense governs which quantities are salient to us, and how it is integrated with fingers involved in the fingers and tool-use. This explains why we represent numbers with hands and body, and it directly informs but does not completely govern how we represent and manipulate quantity with material devices that are not part of the body. I also reviewed how categorizing and abstracting factor into numerical concepts, and noted the term abstract has several meanings affecting how we understand what numbers are and how they emerge. I discussed the topographic organization of the brain, its relevance to numerical structure, and its modulation by visually appreciated features of the material devices used in numbers. I mentioned the involvement of motor-planning activity even when numerical calculation is performed mentally. I also positioned manuovisual access to numerical intuitions as one of two interacting, mutually supporting, and inherently separate systems, the other, of course, being language. How these appear to influence the ways peoples count, as documented in the ethnographic literature, is the topic of the next chapter.

Vandervert, ‘The Origin of Mathematics and Number Sense in the Cerebellum’, p. 4. McPherron et al., ‘Evidence for Stone-Tool-Assisted Consumption of Animal Tissues before 3.39 Million Years Ago at Dikika, Ethiopia’. 86 Overmann and Wynn, ‘Materiality and Human Cognition’, p. 10. 84 85

CHAPTER 5. BEHAVIORAL TRACES To understand the origin of numbers in Mesopotamia, we must start with what other peoples do when they begin counting. For unless and until someone invents time travel, we simply cannot see what ancient peoples did. It’s the same problem we have with their psychological functioning: It’s just not available to us to observe and test in the way it is for extant peoples. The alternatives are limited. Normally, we might try analyzing material forms for their behavioral and psychological implications, like ancient writing implies behaviors such as handwriting and psychological functions such as hand– eye coordination, object recognition, and language. But in this case, there’s no archaeological evidence for us to analyze, because when peoples begin to count, they tend to use things that are archaeologically invisible—they point to things, rearrange the things they’re counting, and count with their fingers or the items themselves or reference sets made of materials that don’t preserve, like wood or string. Sometimes such behaviors leave traces in language, like the words for five that mean hand or the fingers on one hand. But given enough time, processes of linguistic change reduce such terms to unanalyzable syllables, opaque to their original meaning. Sometimes it’s possible to gain insight into earlier states of language. Reconstructing a proto-language means comparing related languages that inherit what they share from an earlier common state. But the languages associated with early writing—Sumerian, Elamite, and Akkadian—are unrelated, and in fact the first two are isolates, languages unrelated to any other known. Since all of these languages are extinct, what we know of them comes from their written remains. But by the time writing was invented, numbers had been around for some time, possibly a very long time. In toto, these circumstances mean we cannot reconstruct the earliest steps into numeracy in the Ancient Near East through archaeological evidence or linguistic means. What we can do is establish that people around the globe count and invent numbers in roughly the same way and then show how the several Mesopotamian number systems conform to the same general pattern. And there is an astonishingly high degree of cross-cultural consistency in numbers. The first number words reflect our perceptual experience of quantity: one and two, often three, and sometimes four, as well as many, typically further specified as big and small. Numbers above the subitizing range are often represented with the hand, with the result that the first terms to emerge from the undifferentiated many are five and ten, amounts later adopted as the number system’s base. Peoples use devices like tallies and knotted strings to extend what the hand can count. By the time peoples are using notations, their numbers have acquired two di65

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mensions—not just accumulating like a tally does, but counting by amounts like ten, productive grouping that forms new numbers in efficient and effective ways. I suggest these qualities reflect the resources and properties of human brains, bodies, behaviors, and material devices in interaction with one another. The first three are speciesuniversal: Our number sense spans the WEIRD/non-WEIRD divide;1 pentadactyl limbs are our physiological norm; and finger-counting and other numerical behaviors are grounded in the primate behavioral repertoire. What limited variability there is in numbers comes from different societies making different choices and combinations in the material forms employed for counting. Even then, there is a high degree of similarity in the materials chosen and combinations used, so that by the time they are written as notations, numbers are organized in only five or six basic patterns.2 This limited variability in numbers starkly contrasts with the wide variability of language, another domain where we argue from what languages do generally to what a particular ancient language would likely have done. Hypothesizing that Mesopotamian peoples invented their numbers in the same way other peoples do is not just the best we can manage, given the limitations on archaeological evidence and linguistic means; it also meets the test of Occam’s razor. Simply, it requires fewer assumptions than arguing for Mesopotamian exceptionality, the idea that Ancient Near Eastern numbers were invented in a manner wholly unique. The argument for Mesopotamian exceptionality rests on three premises, which I review only briefly here and develop in greater detail in the next several chapters. The basic premise is that Ancient Near Eastern peoples were ‘incapable of abstract thought’3 and thus had no abstract concepts—not just of number but of anything else—before they began writing. This premise follows from using Piagetian developmental theory to characterize conceptual change in Ancient Near Eastern societies. Here I will argue that a real phenomenon—the change in behaviors, brains, numerical concepts, and numeracy that occurs when numbers are written—has become associated with inapt and outdated terminology, and that we can gain new insights into that phenomenon when we change the abstract–concrete characterization to something more descriptive. Two other assumptions follow from this premise. First, because it’s assumed there were no abstract numbers, Ancient Near Eastern peoples are thought to have had no number words.4 That is, the numerical lexicon is assumed to have been restricted. Formally, a restricted number system is one in which there are no terms higher than

1 Henrich,

Heine, and Norenzayan, ‘The Weirdest People in the World?’, p. 69. Chrisomalis, Numerical Notation: A Comparative History; Widom and Schlimm, ‘Methodological Reflections on Typologies for Numerical Notations’; Zhang and Norman, ‘A Representational Analysis of Numeration Systems’. 3 Glassner, The Invention of Cuneiform: Writing in Sumer, p. 55. 4 Dahl, ‘Comment on “Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number”’, p. 647; Glassner, The Invention of Cuneiform: Writing in Sumer, p. 55. 2

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twenty,5 though terms can be even more severely limited, reaching no higher than four or five. However, the comparative data associate restricted numbers with emerging number systems and traditional societies, not the kind of large-scale industrialized society Mesopotamia had become long before they wrote their numbers to make them unambiguous to our eyes. Second, the Neolithic tokens are assumed to have been used in one-to-one correspondence with the things they counted.6 The idea is that tokens were so concrete and rudimentary, they must have been the first counting technology used in Mesopotamia. This ignores the relations between tokens,7 something that in and of itself shows they were not used in one-to-one correspondence. The comparative evidence also shows that initial numbers involve technologies far less complex than the tokens, that numbers emerge in societies with far fewer resources to manage than those of the Neolithic period associated with the tokens, and that numerical elaboration both reflects and keeps pace with societal needs for managing complexity. These are complex issues, and it will take the next several chapters to unpack them. In the meantime, I look at the comparative evidence on how peoples begin counting, with the idea that it shows a universal process, making it more likely than not ancient peoples would have followed the same pattern. When you consider crosscultural variability in numbers, keep in mind Chomsky’s observation that a visiting Martian might think all humans speak a single language.8 As I understand his point, once all the surface variability has been stripped away, what remains is universality in how language is produced and processed. If this is true of language—despite the fact that for us Earthlings, the surface variability so easily discarded by the hypothetical visitor from Mars makes languages mutually unintelligible, not just between speakers of different languages, but occasionally between speakers of the same one—how much truer must it be for numbers, where there is no similar ambiguity or opacity in surface form?

A CAVEAT ABOUT THE HISTORIC ETHNOGRAPHIC LITERATURE When I first delved into the historic ethnographic literature to see how different people counted, two things were immediately apparent: Descriptions were remarkably consistent, no matter how widely separated in place and time, and I was hardly the first person to make this assessment. In fact, my impression is that most numerical researchers reach this same general conclusion. Mathematician Øystein Ore, for example, begins his book on number theory by observing that most, if not all, societies count and use their fingers and other material devices of counting, with such material means preceding verbal forms:

Comrie, ‘Numeral Bases’, p. 3. Schmandt-Besserat, Before Writing: From Counting to Cuneiform, p. 6. 7 Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. 8 Chomsky, ‘Language and Nature’. 5 6

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THE MATERIAL ORIGIN OF NUMBERS All the various forms of human culture and human society, even the most rudimentary types, seem to require some concept of number and some process for counting. According to the anthropologists, every people has some terminology for the first numbers, although in the most primitive tribes this may not extend beyond two or three. In a general way one can say that the process of counting consists in matching the objects to be counted with some familiar set of objects like fingers, toes, pebbles, sticks, notches, or the number words. It may be observed that the counting process often goes considerably beyond the existing terms for numerals in the language.9

Some of the anthropological accounts Ore mentions would have been written in the early 20th century. This relative recency means they would reflect the emergence of anthropology as a discipline in its own right, a development heralded by publications such as Edward Tylor’s Primitive Culture (1871) and Anthropology, An Introduction to the Study of Man and Civilization (1881). Subsequent theoretical and methodological advances are perceptible as gradual change, relative to accounts written earlier than the mid19th century, in the education and avocation of ethnographic observers, the kinds of questions asked, and the types of data collected. Other accounts overlap or predate the emergence of a scientific anthropology. These accounts were penned by an eclectic mix of scoundrels, scholars, and Samaritans—explorers, fur trappers, traders, political revolutionaries, escaped convicts, a nobleman or two, several knights, military men, schoolteachers, a few mathematicians, missionaries, clergymen, priests, a bishop later canonized as a saint, and a humanitarian who would win the Nobel Peace Prize—observers who collectively lacked anywhere near the same kind of training and experience characterizing most observers today and whose motivation was something other than collecting reliable ethnographic data. Their writings reflect now-outdated notions of cultural superiority and primitiveness that painfully biased both what they observed and the terms in which they recorded it. Particularly egregious is the writing of Levi Conant, the late 19th-century mathematician who pronounced, ‘those races which are lowest in the scale of civilization, have the feeblest number sense also; or in other words, the least possible power of grasping the abstract idea of number’.10 Conant at least knew his numbers. Few of his predecessors could justly be said to have known much about the numerical practices they recorded. I suspect they just compared the numbers they encountered, often as understood through an unfamiliar language, to what little they knew of their own and remarked on the difference between them. Informants could be unreliable as well. This complicated things like assessing a number system’s extent, something typically performed by asking informants the words for the highest numbers they know. Particular informants might not know the full extent of their society’s number system, or they could simply be making things up, having fun at the expense of the naïve visitor. The latter was apparently the case for the early 19th-century islanders who claimed to have ‘numbers as high as 1,000,000,000,000,000’, a quadrillion, when they were patently inventing silly names, a

9

Ore, Number Theory and Its History, p. 1. Conant, The Number Concept: Its Origin and Development, p. 30.

10

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‘sort of Tonga wit, which is very common with them’.11 Unreliable reports hopefully stand out in some fashion. Mathematical educator Kay Owens suspected this might be the case for Yupno numbers, one of several dozen systems in Papua New Guinea that use the body like a tally, an extended type of finger-counting. As reported,12 Yupno body-counting is atypical: It extends well beyond the range of other such systems, crisscrosses the body an unusual number of times, and includes the penis and testicles, anatomy excluded in other known body-counting systems because of its association with urination and reproduction.13 Yupno body-counting is also isolated linguistically and geographically, as neither peoples speaking related languages nor peoples neighboring the Yupno count on their bodies. While Owens admits the possibility of cultural borrowing, this would make the system’s other atypical characteristics even harder to explain. It surely goes without saying that for scientific research and analyses, reliable data are preferred. The dataset in question—that odd assortment of historic reports from different times and places made by observers of sometimes questionable character consulting occasionally duplicitous informants—is nothing like a scientific sample, one that reflects a designated group with an equal chance of selection. Even the most generous of definitions—the designated group is the entire world, irrespective of time period, cultural definition, and non-independence in the data; randomness involves the rare individual who can read and write encountering something he or she thought worth mentioning—cannot gloss over this fact. Nonetheless, this is the data we have. The dataset is often the only insight into what non-Western numbers were like before they were much influenced by European numerical practices. It is remarkably consistent in what it describes, despite the many vagaries perplexing it. That the data can be so unreliable from a scientific point of view and yet generate the same overall account of how peoples count and use numbers may be surprising. But on consideration, perhaps not: Prehistoric observations of the moon have been consistent, yielding the same months and weeks for peoples who use it to quantify their time, even if the moon turns out not to have been a goddess or made of cheese. Observer unreliability may actually work in favor of data usability: What even the most problematic of instruments can find seems a clear-cut case of what must really be there. Scholars are thus presented with a choice: We can reject the historic data on the grounds of their unreliability, focus on recent data collected through currently approved methods, and gain a synchronic view of numerical variation. Or, we can accept the historic data, contextualize them to mitigate their various biases and flaws, and gain a potential toehold on diachronic change in numbers. I don’t recommend choosing the one and excluding the other; I think we should do both.

11 Martin, An Account of the Natives of the Tonga Islands, in the South Pacific Ocean, Vol. II, pp. 370–371. 12 Wassmann and Dasen, ‘Yupno Number System and Counting’. 13 Owens, ‘The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania’.

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PEOPLES COUNT THE SAME WAY Our perceptual experience means we recognize quantities up to about three without counting and appreciate quantities larger than this range as size differentials, assuming the differences are big enough for us to notice them. These ranges correspond exactly to the first number words to emerge across languages and cultures, even those widely separated by distance and time. The Andaman Islands lie in the Bay of Bengal, south of the Indian subcontinent, and are thought to have been colonized some 60,000 years ago. Tierra del Fuego is the southernmost tip of South America and is estimated to have been inhabited for only 10,000 years. These two locations are separated by about 50,000 years and more than 25,000 kilometers over land. Yet both the Andamanese and Fuegians, at the time contact was recorded, counted to no higher than two or three and indicated higher quantities as many: [The] only numerals in the [Andamans] language are those for denoting “one” and “two,” and … they have absolutely no word to express specifically any higher figures, but indulge in some such vague term as “several,” “many,” “numerous,” “innumerable,”…14 La numération parlée s’arrète pour les Fuégiens au nombre trois; au delà, ils désignent toute collection d’hommes ou de choses par les mots quelques-uns et beaucoup.15 [Spoken numeration stops for the Fuegians at the number three; beyond this, they designate any collection of men or things by the words some and many.]

Such accounts may create the impression that the subitizable quantities are immediately given to us as numbers, but this does not appear to be the case. Ethnographic and linguistic data from Oceania and the Americas show one and two emerging first,16 something that implies both the perceptual salience of subitizable quantities and the appreciation and comparison of quantity in singles and pairs of material objects. Three, though also subitizable, emerges later, not simply as a matter of perceptual salience but often as a combination of two and one. A few behavioral descriptions show three being constructed from the smaller subitizable numbers, like this one recorded in the late 19th century about the Bakaïrí, an indigenous people of northern Brazil: Legte ich 3 Körner …: das Körner-paar wurde zuerst angefasst, … dann links Finger V und IV angefasst und gesagt “ahágé”; das einzelne Korn wurde angefasst, Finger III links zu IV und V herangeschoben, “tokále” gesagt und schliesslich verkündet: “aháge tokále”.17 [I put 3 kernels … the Bakaïrí touched the pair first, then fingers V and IV on their left hand and said “aháge”; then they touched the single grain,

Man, On the Aboriginal Inhabitants of the Andaman Islands, p. 100. Martial, Mission Scientifique du Cap Horn, 1882–1883, Vol. 1, p. 208. 16 Closs, ‘Native American Number Systems’; Eells, ‘Number Systems of the North American Indians’, Parts I and II; Lean, Counting Systems of Papua New Guinea and Oceania. 17 Von den Steinen, Unter den Naturvölker Zentral-Brasiliens. Reiseschilderung und Ergebnisse der Zweiten Schingú-Expedition 1887–1888, p. 494. 14 15

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pushed finger III to join fingers IV and V, said “tokále”, and finally announced: “aháge tokále.”]

The Bakaïrí anchor their understanding of quantity in objects with the quantity of their fingers—not passively viewing the objects, but actively manipulating them, and using the body as a material device to anchor the concept. This suggests numbers are initially the quantities of singles and pairs of objects, experienced visually and manually, understood somatically, and expressed materially. Expression involves recreating the quantity in question with the body or some other material form, or pointing to an aspect of the material environment that exemplifies it. We use this same behavior when communicating numbers between languages: We express quantity clearly and without ambiguity by displaying the requisite number of fingers. Both parties understand the quantity in question by seeing it, whether they also know numbers or share a language with which to express them. This property of visual appreciability will resurface in the later discussion of numerals, for it is something that makes them distinctive among written forms. Using the fingers and, occasionally, the toes, to express numbers and count things is not only common but predominant. Religious scholar Rafael Karsten noted this for the Jibaros and Aguaruna, indigenous peoples of the Marañón River area in northern Peru: The majority of the Jibaros are able to count to “ten”, but only for the five first numerals have they proper names. They always count with the fingers, beginning with those of the left hand, and then also with the toes. The power of counting seems less developed among the Aguarunas; they have proper names only for the three first numerals: chikis, one, hima, two, and mayándi, three and also for five which, like other Jibaros, they denote by the word wéhe amúkei (literally: “I have finished the hand”).18

Ethnologist Buell Quain observed this for the Trumaí people of the upper Xingú River area in central Brazil as well: Enumeration of objects was shown by holding up the appropriate combination of fingers. There were words for the numbers one through ten, but ten also signified “a great many.” Large quantities could be expressed by gesture as well. When Maibu [the chief of the village who acted as Quain’s informant] wanted to indicate many beijú [fish cakes], he held up his hand to their height when stacked.19

Missionary James Barker similarly described the Yanoama, a people of Amazonian Brazil: He pedido hasta 12 objetos, recibiendo la cantidad exacta, mostrándoles 4 dedos de mi mano por tres veces consecutivas. Esto lo comprenden, pero expresar la idea de

18 Karsten, The Head-Hunters of Western Amazonas: The Life and Culture of the Jibaro Indians of Eastern Ecuador and Peru, p. 548. 19 Murphy and Quain, The Trumaí Indians of Central Brazil, p. 77.

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THE MATERIAL ORIGIN OF NUMBERS 11 resulta de difícil comprensión para ellos.20 [I have asked for up to 12 objects, receiving the exact amount, by showing them four fingers of my hand three consecutive times. They understand this, but expressing the idea of 11 is difficult for them to understand.]

The Lanì of West Papua, Indonesia count on their fingers without specific names for the numbers in question: [A Lanì person] does not normally count naming numbers in sequence “one, two, three,” etc. but instead when counting will, for example, say: yi ambìt, yi ambìt, yi ambìt, lambuttogon, kenagan. “This one, and this one, and this one, all together three.” As he does so, he first folds down his little finger in one hand, then his third finger, and then his middle finger, and says, “altogether that is three.”21

Fingers may even be preferred to words, something noted for the Amazonian Tukano people: E ordinariamente preferem indicar o número de dedos (2, 3, etc.) do que proferir os números. De cinco para cima todos, praticamente, se contentam de mostrar o número de dedos correspondentes.22 [And ordinarily they prefer to indicate the number of fingers (2, 3, etc.) than utter the numbers. From five upwards, practically, they are content to show the corresponding number of fingers.]

Counting with the fingers and toes leaves distinctive traces on our numbers. First, the number of our digits becomes the basis for producing new numbers, like ten is the basis of decimal numbers, five and twenty those in quinary and vigesimal systems. Second, the vocabulary for numbers reflects their embodied basis: Terms for numbers like ten mean things like both my hands finished or the upper half (of the body);23 words for hand and man also mean five, ten, and twenty; nouns like digit mean both finger and number, not just in English but in the languages of the Brazilian Aimoré and Hudson Bay Inuit;24 and verbs for counting also mean to finger, seen in the Siberian languages Chukchi and Koryak.25 Number words are often words or phrases describing the fingers and sometimes the toes, as shown by this depiction of Greenland Inuit numbers from Fridtjof Nansen, the 19th-century explorer, scientist, and humanitarian awarded the Nobel Peace Prize in 1922: [The Eskimos] count upon their fingers: One, atausek; two, mardluk; three, pingasut; four, sisamet; five, tatdlimat, the last having probably been the original word for the

20 Barker,

‘Memoria Sobre la Cultura de los Guaika’, p. 487. Larson, ‘Western Dani (Lani), Indonesia’. 22 Da Silva, The Indigenous Civilization of the Uaupés, p. 257. 23 Closs, ‘Native American Number Systems’; Dixon and Kroeber, ‘Numeral Systems of the Languages of California’. 24 Conant, The Number Concept: Its Origin and Development; Richardson, ‘Digital Reckoning among the Ancients’. 25 Antropova and Kuznetsova, ‘The Chukchi’, p. 800. 21

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hand. When an Eskimo wants to count beyond five, he expresses six by saying “the first finger of the second hand” (arfinek or igluane atausek); for seven he says “the second finger of the other hand” (arfinek mardluk), and so forth. When he reaches ten he has no more hands to count with, and must have recourse to his feet. Twelve, accordingly, is represented by “two toes upon the one foot” (arkanek mardluk), and so forth; seventeen by “two toes on the second foot” (arfersanek mardluk), and so forth. Thus he manages to mount to twenty, which he calls a whole man (inuk nâvdlugo).26

Though we all count on our fingers, and some of us, on our toes, this does not mean we are fated to count by fives, tens, and twenties.27 Counting with the fingers without developing numbers based on multiples of five might use the segments of the fingers, counting to twelve if the thumb is omitted and fourteen if the thumb is included.28 Another example is found with the Yuki, a now-extinct people of Northern California. Their method of counting involved placing sticks in the spaces between the fingers, rather than using the fingers themselves; in each interstice, ‘when the manipulation was possible, two twigs were laid’.29 This practice yielded a number system based on four and eight. The number words associated with four Yuki number systems (Table 5.1) show that counting with material devices produces numbers in a manner highly similar, if not identical, to using the fingers as the basis, as was just shown with the Greenland Inuit. The linguistic analysis offered by anthropologists Alfred Kroeber and Roland Dixon30 suggests that four Yuki groups—the Round Valley, Huchnom, Coast, and Wappo Yuki in Northern California—counted with interstitial sticks. This shared behavior with a common material form, however, did not yield similar numerical outcomes, demonstrating both that we are not fated to count by fives and that even cultures with significant linguistic and cultural affinities need not construct their numbers the same way. As shown in Table 5.1, the words for one and two are clearly related in all four dialects. Three is clearly related in the Round Valley, Huchnom, and Coast Yuki dialects, and four is constructed similarly as two of something related to sticks or fingers. In the southern dialect, Wappo, three is unrelated, and four is unanalyzable. These data suggest the words for the subitizable quantities developed before the Yuki split into four groups, with the Wappo perhaps splitting off earlier than the rest. The numbers higher than four differ substantially in their construction, to the extent that the systems are even organized differently: Round Valley numbers are an octal, or four- and eight-based system, Huchnom numbers are based on five and twenty, and the remaining two are based on five and ten.31

Nansen, Eskimo Life, pp. 194–195. Dixon and Kroeber, ‘Numeral Systems of the Languages of California’, p. 668. 28 Huylebrouck, ‘Tellen Op de Handen in Afrika en de Oorsprong van het Duodecimale Systeem’. 29 Kroeber, Handbook of the Indians of California, p. 176. 30 Dixon and Kroeber, ‘Numeral Systems of the Languages of California’, p. 685. 31 Closs, ‘Native American Number Systems’, p. 4. 26 27

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Nr

Round Valley (Octal)

Huchnom (5–20)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

pan-wi op-i molm-i o-mahant hui-ko mikas-tcil-ki mikas-ko paum-pat hutcam-panwi-pan hutcam-opi-sul molmi-sul omahant-sul huiko-sul mikastcilki-sul mikasko-sul hui-co(t) panwi-hui-luk opi-hui-luk molmi-hui poi omahant-hui-poi

pu-we op-e molm-e kes-ope pu-putc pu-tal opi-nun kina-sa-nun helpiso-pu-tal helpiso-humate helpiso-pu-tik helpiso-ope-tik helpiso-molme-tik a’la-pu-tan a’lau’ a’la-pu-tik a’la’-ope-tik a’la-kinasanun-tik pu-al-yak pu-tan pu-al-yak

Coast (5–10) pow-ik op-ik molm-ik hilkil-opik pou-pat pou-tit ope-tot molme-tit hikil ope-tit popate-tit

Wappo (5–10) pawe hopi hoboka ola gada pa-tenauk hopi-tenauk hopi-han paw-alak mahaic mahaic-bawa-len mahaic-hopi-len mahaic-ola-len mahaic-patenauk-len

op-keckeneclak

mahaic-paalak-len hopi-hol

Table 5.1. Number words in four Yuki dialects. The numbers four and eight in Round Valley Yukian and five and ten in the other systems, shown in bold, reflect various configurations of sticks, fingers, and hands. These numbers were also productive, meaning they served as the basis for forming other, usually higher, numbers. For the main Yuki dialect, Kroeber noted that every word from four up into several hundreds described the counting process; he glossed four to twenty as ‘two-forks, middle-in, even-chilki, even-in, one-flat, beyond-one-hang, beyond-two-body, three-body, two-forks-body, middle-inbody, even-chilki-body, even-in-body, middle-none, one-middle-project, two-middleproject, three-middle-project, two-forks-middle-project’.32 Similar constructions in the other dialects suggest they too used sticks for counting: In Huchnom, the word for twenty meant ‘1-stick-stand’, and in Coast Yuki, the words for the hundreds (100: pu-al; 200: ope-al) referred to sticks, as did the word for nine in the Wappo dialect.33 Counting with interstitial sticks influenced the Round Valley numbers toward octal organization, while the other three systems were decimal variants,34 something plausibly related to the amount of inter-group contact experienced by the different groups. Data compiled from Dixon and Kroeber, ‘Numeral Systems of the Languages of California’, p. 677.

These data suggest all four groups independently developed their numbers four and higher, though the similar terms for hundreds in the Huchnom and Coast Yuki systems make it likely these groups were in later contact. The distinctive octal organization of the Round Valley system implies these numbers developed in relative isolation from the other groups. This interpretation is supported by evidence that the Round Valley group fought frequently with other Yuki groups, especially the Huchnom, as well as Kroeber, Handbook of the Indians of California, p. 177. Dixon and Kroeber, ‘Numeral Systems of the Languages of California’, p. 685. 34 Closs, ‘Native American Number Systems’, p. 4. 32 33

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non-Yuki groups.35 The remaining systems were likely influenced toward decimal organization through contact with outside groups, most of whom had systems based on five, ten, and twenty. This interpretation is supported by cultural affinities between the Huchnom and Pomo, a group with quinary–vigesimal and decimal number systems,36 as well as Wappo linguistic characteristics implying significant amounts of both time after diverging from the main Yuki group and cultural influence from surrounding non-Yuki groups.37 As the Yuki number words show, early number words express quantity by describing a material form exemplifying it. That is, these number words point to the exemplar. This is true of even the earliest number words, as missionary Martin Dobrizhoffer noted for the Abipón people, a now-assimilated indigenous group of Argentina’s Gran Chaco region. Essentially, number words function as verbal gestures rather than physical ones, though physical gestures that display the fingers can accompany them, especially if the words relate to the fingers: They make up [numbers higher than three] by various arts: thus, Geyenk ñatè, the fingers of an emu, which, as it has three in front and one turned back, are four, serves to express that number. Neènhalek, a beautiful skin spotted with five different colours, is used to signify the number five. If you interrogate an Abipon respecting the number of any thing, he will stick up his fingers, and say, leyer iri, so many. If it be of importance to convey an accurate idea of the number of the thing, he will display the fingers of both hands or feet, and if all these are not sufficient, show them over and over again till they equal the number required. Hence Hanámbegem, the fingers of one hand mean five; Lanám rihegem, the fingers of both hands, ten; Lanám rihegem, cat gracherhaka anámichirihegèm, the fingers of both hands and both feet, twenty.38

This was also true of the Andamanese: To express “one,” they hold up the forefinger of either hand and utter the word ū·ba-tū·l- or ū·ba-dō·ga-; to denote “two” they hold up the first two fingers and say īkrô·r-.39

Number words can function iconically as well, representing quantity by recreating it with sound. This is the case for the Mundurukú of Brazil, whose words for one, two, three, and four have a quantity of syllables that matches the quantity they express, so they represent not just by pointing verbally at material exemplars of quantity—fat, arms, and parents—but by recreating it with sound.40 Mundurukú number words also demonstrate compounding, as the word for two, xep-xep, is the word fat said twice or reduplicat-

Golla, California Indian Languages, pp. 188–193. Dixon and Kroeber, ‘Numeral Systems of the Languages of California’, p. 685. 37 Kroeber, Handbook of the Indians of California, pp. 203–217. 38 Dobrizhoffer, An Account of the Abipones, an Equestrian People of Paraguay, Vol. 2, p. 169. 39 Man, On the Aboriginal Inhabitants of the Andaman Islands, p. 32. 40 Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’. 35 36

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ed; the word for about three, eba-pûg, means ‘your arm(s) and one’, since pûg is the word for one; and the word for about four, eba-dip-dip, means ‘your parent(s)’ reduplicated.41 When people don’t have words for specific numbers, and perhaps lack concepts of even the smallest subitizable quantities, they employ different strategies, like memory for where objects are located. This has been demonstrated by cognitive neuropsychologist Brian Butterworth and his colleagues. They compared how children from societies with restricted or unrestricted numbers recreate arrays of objects; the former remember where the objects are located, while the latter enumerate them.42 People also recognize individuals—social others; valuable animals—as individuals, something that may obviate the need to count them. Missionary Louis Schulze observed this was the case for the Australian Arrernte people: Although [the Arrernte are] only able to count up to four, they can individualise large herds of cattle or horses, and can tell at once if one is missing, and which, and what its color or appearance may be.43

Similarly, the Nuer, a people of the South Sudan and southwestern Ethiopia, ‘know each of their animals and treat them as individuals but do not count them as a herd’.44 Since the Nuer were also described as counting the teeth of their cattle,45 not counting them as individuals may reflect a cultural practice, knowing other animate beings as individuals rather than counting them, retained after numbers became available. Such would be consistent with the idea that what numbers are as concepts is informed by behaviors, habits, and social knowledge, as was discussed in conjunction with Hutchins’ model of conceptual blending in Chapter 3. When people don’t have number concepts or words, numbers are nonetheless usable and intelligible, made tangible and communicable by material means. This is known as one-to-one correspondence, a behavior with material forms with the potential to scaffold concepts of discrete quantity. An example comes from the memoir of William Buckley, a transported convict who escaped prison to live with the aboriginal Wathaurung people of southern Australia. Their highest number counted, as estimated some 40 years later, was ten,46 suggesting it could have been lower at the time Buckley made his observation: Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’. Butterworth and Reeve, ‘Verbal Counting and Spatial Strategies in Numerical Tasks: Evidence from Indigenous Australia’; Butterworth, Reeve, and Reynolds, ‘Using Mental Representations of Space When Words Are Unavailable: Studies of Enumeration and Arithmetic in Indigenous Australia’. 43 Schulze, ‘Aborigines of the Upper and Middle Finke River: Their Habits and Customs, with Introductory Notes on the Physical and Natural-History Features of the Country’, p. 220. 44 Schmandt-Besserat, ‘The Emergence of Recording’, p. 847, citing Evans-Pritchard. 45 Evans-Pritchard, The Nuer: A Description of the Modes of Livelihood and Political Institutions of a Nilotic People, p. 22. 46 Matthews, referenced in Blake, Clark, and Krishna-Pillay, ‘Wathawurrung and the Colac Language of Southern Victoria’. 41 42

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[A] messenger, came to us; he had his arms striped with red clay, to denote the number of days it would take us to reach the tribe he came from…. The time stated for this march would be fourteen days.47

Forty years later, there was still no word in the Wathaurung language to express a number as high as fourteen. This, however, was no barrier to their using such numbers. The messenger added a stripe to his arm each day of his journey; going home, he erased one a day. His use of numbers in material form helped him navigate the Australian outback, a harsh and potentially deadly terrain. Nor is this the only way bodies are used for counting, as there are a few dozen number systems in Papua New Guinea where ordinal sequences are counted out using positions on the body, typically starting with the fingers, proceeding up the arm over the head and back down the other side to count to numbers as few as twelve or as high as seventy-four, but typically counting to between twenty-three and twenty-seven.48 Unsurprisingly, given cross-cultural proscriptions limiting public touch and behavioral imitation, especially for prestige bodies and behaviors, such systems may be restricted to use by adult men.49 A similar behavior is mentioned frequently in descriptions of how peoples with few numbers coordinate celebrations and trade: They tie knots in strings or tear leaves to represent the number of days that must elapse before the desired date. Here is a description from the Kwoma people of New Guinea, the large island divided between the independent state of Papua New Guinea and the Indonesian province West Papua: Word had been sent announcing the trading by a mnemonic device consisting of a knotted string—one knot for each day which was to intervene between the reception of the string and the actual trading.50

Similarly, from the southeastern part of Bougainville Island, Papua New Guinea, the Siuai people use branches with the requisite number of leaves: [The] Siuai have learned to express [whole days as units with] more precision, with the aid of a tally (mitamita). In setting a date for, say, a feast, the host sends to each of his principal guests a palm frond having a number of leaves equal to the number of days before the feast. Thereafter, the host and his guests tear off one leaf each day to mark the passage of time before the feast.51

The Shona, an indigenous people of Zimbabwe, Botswana, and southern Mozambique, use the same strategy in a coordinating the price of a bride:

47 Morgan, The Life and Adventures of William Buckley: Thirty-Two Years a Wanderer amongst the Aborigines of the Then Unexplored Country around Port Phillip, Now the Province of Victoria, pp. 48–49. 48 Lean, ‘Counting Systems of Papua New Guinea and Oceania’; Owens, ‘The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania’. 49 Wassmann and Dasen, ‘Yupno Number System and Counting’. 50 Whiting, Kwoma Journal, p. 43. 51 Oliver, A Solomon Island Society: Kinship and Leadership among the Siuai of Bougainville, p. 98.

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THE MATERIAL ORIGIN OF NUMBERS [The suitor presents] a stick of fibre, knotted several times, to invite the girl’s father to come on a certain day to inspect the cattle he is to receive. He gives … the piece of fibre, called mukosi, to the father. Each knot represents a day and it is intended to show him the number of days that must pass before the date on which he is expected.52

Recipients untie one knot or cut off one section of leaf daily until none are left, indicating the arrival of the specified day. These are examples of what I call quantification behaviors, behaviors with numbers that don’t depend on the availability of explicit concepts or words but which function nonetheless to represent and manipulate numbers in the absence of these things. These methods of keeping track of days by means of stripes, knots, and tears exemplify one-dimensional devices for representing quantity (Fig. 5.1). Other onedimensional devices include the tally, a stick or bone marked with series of notches; marks, stripes or dots made or painted on the ground, body, or other surface, and stones, pebbles, sticks, and beans, small objects accumulated in piles. The identity of the object being enumerated is either remembered or known contextually. At first glance, stripes, marks, dots, knots, notches, stones, beans, pebbles, sticks, leaves, and tears may not seem particularly similar in form, but they all indicate quantity between two sets of objects in one-to-one correspondence: One indicator of quantity means one of whatever it is being counted. One-dimensional devices also have in common the ability to scaffold concepts of, and words for, higher quantities than what the fingers and toes typically represent. When number words emerge to accompany such devices, these are very likely to be an ordinal sequence, or a sequence whose order indicates successively higher quantity but not the size of the interval between numbers. A good example of this are the numbers of the Oksapmin people of Papua New Guinea, who use the human body as a tally. As extensively documented by psychologist Geoffrey Saxe, the Oksapmin count from one to twenty-seven using body positions. These start from the right thumb, proceed across the right hand to the little finger, go up the right arm through the wrist, forearm, inner elbow, biceps, shoulder, neck, ear, and eye, and then cross the midline of the body at the nose. The sequence is then repeated in reverse across the left side of the body, except it is reversed again across the left hand to end with the little finger.53 Oksapmin numbers are named for the body part referenced by touching it, mirroring the way other groups construct number words based on fingers and material devices like sticks, except the sequence is not organized around a productive base like five or ten.

52 Gelfand, The Spiritual Beliefs of the Shona: A Study Based on Field Work among the East-Central Shona, p. 48. 53 Saxe, Cultural Development of Mathematical Ideas.

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Fig. 5.1. One-dimensional devices. These artifacts for representing quantity are similar in accumulating quantity without grouping it. While their use does not depend on the availability of concepts and words for numbers, they can act as scaffolds for concepts of higher numbers, as well as potential relations such as more, less, not equal, between, count, and possibly plus-one. Most of these would not leave traces in the archaeological record, though finger-counting in particular influences the structure and organization of number words and number concepts. Compare with the two-dimensional devices in Fig. 5.2. Images from the public domain.

Tallies can be constructed in various ways, including many that would be invisible in the archaeological record. In Lesu, a village on the island of New Ireland in Papua New Guinea, people celebrate births with a communal feast. Preparing for the feast involves collecting and counting food items: Bananas are taken off the stem by the men and arranged in another pile, and the bundles of fish, pig, and turtle are also opened. Each woman puts on the ground either her basket or her mat, which form a long row on the ground between the pile of taro and the pile of bananas. The infant’s maternal grandfather then counts a blade of grass for each woman present, putting one blade on each basket and mat to see if the latter correspond to the number of blades of grass he has, which represent the number of women. This is the method of counting.54

In Polynesia, the Māori of New Zealand created a tally from the beaks of birds prepared for consumption, modifying the objects to have a feature used later for their counting: The bulk of the birds from the snaring are set aside for preserving. They are plucked and the bones removed, leaving the lower beak remaining with the flesh, for when 54

Powdermaker, Life in Lesu: The Study of a Melanesian Society in New Ireland, p. 75.

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THE MATERIAL ORIGIN OF NUMBERS the prepared birds of each person are counted it is by means of these beaks that the tally is made.55

In Tikopia, geographically located in the Solomon Islands but culturally Polynesian, “The count of the sacred yam” was made, by a simple but ingenious method. To ascertain accurately how many hillocks had been planted, a man took a piece of coconut leaf and going round the plot tore off pinnules [divisions of a leaf] one after the other and threw one on each hillock. When each had its pinnule he collected these and counted them. This obviated the risk of missing any hillocks, as might have easily occurred if they were counted by eye alone.56

For the items being counted and the devices used to represent the count, the correspondence remains one to one. In the process of accumulating the count, a device is created that has the potential to represent the items counted in their absence. Fortunately for archaeologists, tallies can also be created of more durable stuff: For the Orokaiva people in Papua New Guinea, ‘the village constable who was in charge of the plantation was almost invariably illiterate, [and] some of them kept an “attendance stick” for each man. A notch was cut in the stick for each day of absence other than that caused by illness or the death of a close relative’.57 More likely than not, the marks on the attendance stick tally were ungrouped, since there is no organic grouping to them in the way that can occur naturally when we count on our hands and feet: If it is necessary to continue counting [beyond ten] the Jibaro seizes the toes of one foot, one by one, and counts chikichi, himera, menéindu (one, two, three), etc. When he arrives at the fifth toe he says: huini náwi amúkahei, “here I have finished one foot” (Huini = here, náwi = foot), it being understood that he has begun with the hands. The said phrase, therefore, is equal to “fifteen”.58

Tallies can be grouped, as shown by the familiar tally marks used today: ༛. Since marks are easy to add but more difficult to change or remove once they have been made, grouping must be implemented as part of the process of making marks, something implying forethought. Grouping by fives, tens, and twenties is plausibly informed by counting on the body, since such amounts (Fig. 5.2) are not naturally inherent in any of the objects we might count, nor are they a function of our perceptual system for quantity.

Firth, Economics of the New Zealand Māori, p. 164. Firth, The Work of the Gods in Tikopia, pp. 186–187. 57 Crocombe, Communal Cash Cropping among the Orokaiva, p. 15. 58 Karsten, The Head-Hunters of Western Amazonas, p. 548. 55 56

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Fig. 5.2. Two-dimensional devices. These include the natural grouping of the body, shown on the left, something that influences devices that are not part of the body to be grouped by fives, tens, and twenties. Grouping adds a second dimension to numbers: The tokens used in Mesopotamia on the right not only accumulate (shown along the horizontal dimension), they also group (the vertical dimension), adding correspondences between the items in the reference set: Ten of the small cones on the bottom row equal one small sphere on the second row, six small spheres equal one large cone on the third row, and ten large cones equal one cone marked with a sphere on the top row. Potential relations include those of one-dimensional devices, plus those of grouping. Intermediate between fingers and tokens are reference sets that are not part of the body and that correspond to the objects being enumerated in amounts other than one to one. Images from the public domain.

Material devices other than the body can be used to count objects in a way that both accumulates and draws upon the grouping influenced by the body. For example, people in the Marshall Islands used stones to count groups of ten coconuts: ‘er legt für zehn Kokosnüsse einen Stein als Marke hin’59 [he uses a stone as a mark for ten coconuts]. For the coconuts being counted and the stones used to represent the count, the correspondence is ten to one rather than one to one. In the process of counting, a device is created—the accumulation of stones—that has the potential to represent the items counted in their absence. The New Zealand Māori used the objects being counted as a tally: Sometimes when counting a number of objects the Māoris would put aside 1 to represent each 10, and then those so set aside would afterward be counted to ascertain the number of tens in the heap. Early observers among this people, seeing them count 10 and then set aside 1, at the same time pronouncing the word tekau, imagined that this word meant 11, and that the ignorant savage was making use of this number as his base. This misconception found its way into the early New Zealand 59 Krämer, Hawaii, Ostmikronesien und Samoa: Meine Zweite Südseereise (1897–1899) zum Studium der Atolle und Ihrer Bewohner, p. 438, footnote 1.

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THE MATERIAL ORIGIN OF NUMBERS dictionary, but was corrected in later editions. It is here mentioned only because of the wide diffusion of the error, and the interest it has always excited.60

Conant noted receiving the information from William Leonard Williams, the son of the author of the dictionary in question. Its 1944 edition stated, ‘The Native mode of counting is by elevens, till they arrive at the tenth eleven, which is their hundred; then onwards to the tenth hundred, which is their thousand’, to which a footnote added, ‘This seems to be on the principle of putting aside one to every ten as a tally’.61 It is indeed possible to use every eleventh item to mark the previous ten, and then to count the markers in the same fashion, and this would yield the base eleven number equivalent to a decimal count. However, there is no evidence to support the assertion that Māori numbers were extensively developed using eleven as their base, with ‘simple words for 121 and 1331, i.e. for the square and cube of 11’,62 since both Williams’ dictionary and Lee’s 1820 grammar63 show Māori number words as decidedly decimal. Māori counting appears to have been thoroughly misunderstood, or possibly represents an instance of informants having fun at an observer’s expense;64 most likely, the Māori were simply putting aside every tenth item to create a tally, a highly effective method of counting. Grouping by twenty occurs when counting is extended to the hands and toes of more than one man. For the Greenland Inuit in the late 19 th century, twenty-one was ‘“one on the second man” (inûp áipagssâne atausek). Thirty-eight [was] expressed by “three toes on the second man’s second foot” (inûp áipagssâne arfinek pingasut), forty by “the whole of the second man” (inûp áipagssâ nâvdlugo), and so forth’.65 Perhaps nowhere is counting by the digits of multiple men, ten rather than twenty in this case, so extensively codified as that reported for the ceremonial counting of the Tonga of Polynesia: Presentations of food, etc., are counted in the presence of the assemblage. Baskets of food are set out in orderly rows, and a man goes along the rows, touching and counting the baskets…. The counter calls aloud: “One, two, three,” and so on up to ten, when a second man rises, holding a staff, and calls: “One.” Then the counter goes on with the second ten, “One, two, three,” and so on till the second ten is completed, when the man with the staff calls: “Two.” So it goes on, the first man counting the individuals in groups of ten, the second man keeping the tally of tens, till one hundred, ten tens, is reached, when the second man calls “One hundred,” and a third man jumps up and calls “One.” The second hundred is counted in the Conant, The Number Concept: Its Origin and Development, p. 123. Williams, A Dictionary of the New-Zealand Language, and a Concise Grammar; to Which Are Added a Selection of Colloquial Sentences, p. xv. 62 Conant, The Number Concept: Its Origin and Development, p. 123; also see Balbi, ‘Observations sur la Classification des Langues Océanniennes’, pp. 256–257. 63 Lee, A Grammar and Vocabulary of the Language of New Zealand. 64 Craik, The New Zealanders, pp. 416–417. 65 Nansen, Eskimo Life, p. 195. 60 61

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same way as the first by the same two men, and when it is completed the third man, who has been silent meanwhile, calls “Two.” If the number be so great that the tenth hundred is completed, the third man, he who keeps the tally of hundreds, calls “One thousand,” and a fourth man joins in, and says “One,” and it will be his task to keep the tally of the thousands.66

Given Tonga’s proximity to, and cultural interaction with, Fiji, it is perhaps unsurprising that such collaborative counting is found there as well: The Fijian verb bunuca meant ‘to tally, or count the number of tens while another is counting the units, as in counting yams [and other types of items]’.67 Collaboration makes counting easier by reducing the demands on individual working memory: Counting becomes a simple matter of repetitively accumulating to ten, and the involvement of new individuals keeps track of points where units become tens, tens become hundreds, and so on. As each participant is also free to use his fingers, it is really just finger-counting on the hands of multiple people. Such collaboration also mitigates issues of public touch and behavioral imitation found in the body-counting systems of Papua New Guinea. Counting by groups can involve other material forms besides the body, as shown by the Warao, an indigenous people of the Orinoco Delta in Venezuela: Si acaso tienen que contar hasta ochenta, hasta cien, para no equivocarse, depués de veinte, después de “un guarao”, hacen una raya en el suelo. Así pues: — Una raya son veinte, un guarao, los dedos de manos y pies de un guarao. — Dos rayas, cuarenta, los dedos de pies y manos de dos indios. — Tres rayas, sesenta. — Cuatro rayas, ochenta. — Cinco rayas, cien, los dedos de manos y pies de cinco indios.68 [If perhaps they have to count to eighty, to a hundred, in order not to be wrong, after twenty, after “one Guarao,” they make a line on the ground. So that: — A line is twenty, one Guarao, the fingers and toes of one Guarao. — Two lines, forty, the fingers and toes of two Indians. — Three lines, sixty. — Four lines, eighty. — Five lines, one hundred, the fingers and toes of five Indians.]

Grouping need not be confined to the devices used for counting. A practice common throughout Polynesia is grouping the objects being counted in pairs or fours. The ob-

66 Collocott, ‘Supplementary Tongan Vocabulary; Also Notes on Measuring and Counting, Proverbial Expressions and Phases of the Moon’, pp. 210–211. 67 Hazlewood, A Feejeean and English Dictionary: With Examples of Common and Peculiar Modes of Expression, and Uses of Words, p. 20. 68 Turrado Moreno, Etnografía de los Indios Guaraúnos, p. 203.

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jects enumerated this way tend to be items that are both socially valuable and counted often.69 In the Marquesas, for example, [Breadfruit] were gathered up and counted in bunches of fours called pona (literally, knot). Thus a tahi pona was one bunch of four, [an] iva pona, nine bunches, or thirtysix breadfruit. The whole system of counting breadfruit differed from the normal count.70

This behavior plausibly reflects the desire to reduce the labor involved in counting hundreds or thousands of things like coconuts by handling them: Counting by pairs cuts the number needing to be counted in half, while counting by fours cuts it to a fourth: …most of the objects which the natives have occasion to enumerate, being articles of food, and of small size (such as yams, cocoa-nuts, fish, and the like), can be most conveniently and expeditiously counted in pairs. This mode is therefore universally adopted. Taking one in each hand, the native, as he throws them into the storehouse, or on to the heap, counts one; for two pairs, he says two; for ten pairs simply ten, and so on. Hence each number has a twofold value, one for objects counted singly, and one for those reckoned in pairs.71

As Conant speculated was the case for the Māori—though wrongfully, as it turned out, for their purported counting by elevens—such practices have an effect on number words. Terms that are productive, meaning they serve as the basis for forming new words, usually for higher numbers, emerge at points that reflect the material grouping employed. In decimal systems, productive words emerge at ten, hundred, thousand, and so on, reflecting grouping in the device used for counting, the fingers of the two hands; in octal, as the Yuki example demonstrated, productive words are four and eight, representing groupings of interstitial sticks. In Polynesian number systems that count items by twos and fours and eights, productive terms emerge at ten of the base unit: ten twos,72 ten fours,73 and in Mangareva, ten eights.74 By no means do I wish to suggest either that there is a single path to numbers or that the decimal, plus-one linear structure common to many numerical traditions and dominant in the West is their only basis. In Sora, a language of eastern India, numbers are vigesimal with a sub-base of 12, so the word for nineteen means twelve and seven, the 69 Beller and Bender, ‘The Cognitive Advantages of Counting Specifically: An Analysis of Polynesian Number Systems’; Bender and Beller, ‘Extending the Limits of Counting in Oceania: Adapting Tools for Numerical Cognition to Cultural Needs’. 70 Handy, The Native Culture in the Marquesas, p. 184. 71 Hale, United States Exploring Expedition. Vol. VI. Ethnography and Philology, p. 247. 72 Bender and Beller, ‘“Fanciful” or Genuine? Bases and High Numerals in Polynesian Number Systems’. 73 Campbell, A Voyage Round the World, from 1806 to 1812; in Which Japan, Kamschatka, the Aleutian Islands, and the Sandwich Islands Were Visited. 74 Bender and Beller, ‘Extending the Limits of Counting in Oceania’.

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word for thirty-three means twenty and twelve and one, and the word for forty means two twenties.75 In Kakoli, a language of the Upper Kaugel Valley in the Southern Highlands of Papua New Guinea, numbers are organized as a base-24 system counted in sets of four. Each new set is considered to be a part of set that follows, rather than an addition to the just-completed set, so the word for eighteen means two of twenty,76 understood as four complete sets of four (or sixteen) and two of the four in the fifth set. When a cycle of 24 has been completed, counting continues to 32, stops to recognize a completed cycle of 24 plus a remainder of eight, and resumes counting the next cycle of 24 by starting with the eight left over from the previous cycle. By no means does this grouped counting preclude the use of tallies, as they are an occasional feature of Kaugel counting, 77 but it does imply significant manipulation of material forms—objects organized by fours and twenty-fours. Sora and Kakoli numbers also demonstrate the considerable influence material grouping has on numerical structure and organization. This is not surprising, as it is no more than the influence that finger-counting and devices like tallies have on characteristics like productive grouping and linearity. What is perhaps surprising is the variability that can result from the different ways societies use and organize material forms to represent and manipulate numbers. To someone like me accustomed to bland incremental decimality, the resultant structures can be strange, counterintuitive, astonishingly beautiful, and wonderfully illuminative of the human capacity for number.

QUESTIONS WE MIGHT ASK AT THIS POINT At this point, the astute reader might be asking questions like, why do we use material devices for counting? As the ethnographic data suggest, from the very beginning of numbers, material devices make our innate sense of quantity tangible, which in turn lets us express and manipulate it. This is a critical role because using material devices to represent and manipulate quantity is how number concepts are realized and number words occasioned. Given concepts and number words, even as simple a thing as counting is more difficult without touching the enumerated objects or using some kind of device to represent the count. Von den Steinen noted this for the Bakaïrí, who both touched the objects he asked them to count and used their fingers to represent the number.78 Counting without touching or manipulating the objects being enumerated is difficult for quantities greater than about four, as moderated by the degree to which the objects are visually distinguishable and individualizable; this is a function of how our perceptual system for quantity works. Counting without using some kind of device to represent the count is difficult because it represents a greater demand on cognitive resources like working memory and attention. Arguably, such difficulties persist even when number concepts and number words are available, since concepts and words do not change our perceptual experience of quantity, nor increase the capacity of our working memory or attention. Anderson and Harrison, ‘Sora’, pp. 318–319. Bowers and Lepi, ‘Kaugel Valley Systems of Reckoning’, p. 313. 77 Bowers and Lepi, p. 314. 78 Von den Steinen, Unter den Naturvölker Zentral-Brasiliens, p. 505. 75 76

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Another question might be, what does materiality actually do in numbers? The ethnographic data show that material forms act as visual and manual stimuli for our perceptual system for quantity. They interact with cognitive resources like working memory and attention, expanding our capacities well beyond what our brains can accomplish on their own. Their ability to represent and manipulate numbers enables us to fulfill social needs and purposes through them, while revealing combinations and patterns that we can exploit further. As they exemplify and represent quantity, material devices influence the way in which we group quantity productively. This is particularly true of the fingers and toes, whose use in counting causes our numbers to be grouped in multiples of five. Material forms, usually but not exclusively ones that are not part of the body, scaffold concepts of numbers higher than what we can subitize or represent with fingers and toes. This scaffolding not only helps us explicate the relations between numbers, it codifies number knowledge in artifactual form, thereby distributing it between individuals and generations. Material forms occasion and structure naming in language. The availability of number words, in turn, opens up our numerical conceptualization to further expression and elaboration. These contributions reveal, as was discussed in Chapter 2, material forms as integral to our numerical cognition, agentive in influencing our numerical content, organization, and structure, and meaningful through what they are and how we use them. Why are the material devices presented in the order chosen? First, it’s reasonable to think that numbers begin in an environment that contains nothing but the perceptual experience of quantity, language, and material culture. More strongly, it would be unreasonable to believe otherwise, since this would entail including a number of rather odd assumptions. Sequencing the use of the fingers next is inferred from both the neurological underpinnings of finger-counting, reviewed in Chapter 4, and the fact that so many languages in Oceania and the Americas contain numbers with unambiguous etymological roots in finger-counting. Since perceptual experience and finger-counting precede other forms of materially mediated counting, these logically follow them in the sequence. In this case, it is reasonable to assume that one-dimensional counting precedes two-dimensional counting. This inference will be supported using the case of Mesopotamia, where there is archaeological evidence of possible one-dimensional tallies that occurs much earlier than the two-dimensional tokens and notations. But as the ethnographic data just reviewed also shows, there is often an implicit two-dimensionality to counting with the body, if the body is used as a grouped material structure, though the Oksapmin ordinal numbers also show this need not always be the case. Why are most of these cultures located in Oceania and the Americas, and what about nonindependence in the data due to cultural and linguistic affinities? The concentration of so-called emerging number systems in Oceania and the Americas, I will argue in the next chapter, is an effect of residential recency, since these are the continents most recently populated by the migration of ancient peoples. There are also demographic factors that influence whether numbers are expressed, including group size and the amount of contact between groups. Cumulatively, these say something about the relation between a society and its material culture, insights important to understanding the role of the latter in our cognition. Non-independence of the data is always a concern in crosscultural research, especially whenever a sample is small, as it is for historical number systems. Here I have tried to mitigate any effects of non-independence by presenting,

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when possible, similar examples from both sides of the continental peopling; while all Polynesian and all South American peoples are related to some degree both behaviorally and linguistically, societies in Polynesia are less related to those in South America, and vice versa. Nonetheless, observations of related peoples help us, at least to some extent, verify the general accuracy of the reports. They also reveal important differences, since even societies with high cultural and linguistic affiliation can vary in how they use the same material forms for representing and manipulating numbers, as the Yuki example showed. Why are some number words, usually the smallest ones, so much shorter than others? In fact, it is not just that these words are shorter, they are often considered more proper or more real or special in some way. We see this same judgment rendered for many of the historic number systems: Abipón numbers: ‘The Abipones can only express three numbers in proper words. Iñitára, one. Iñoaka, two. Iñoaka yekainì, three’.79 Yuki numbers: ‘The Yuki managed their count with only three real numeral words: panwi, one; opi, two; molmi, three’.80 Greenland Inuit numbers: ‘The Eskimo language, like most primitive idioms, has a very undeveloped system of numerals, five being the highest number for which they have a special word’.81 It was also noted for the Jibaros and Aguarunas.82 Arguably, what is occurring is that these proper names are nothing more than longer ones whose origin is now invisible: Etymological transparency ‘tends to be obscured over time and/or through frequent use via the processes of lexicalization that affect linguistic expressions in general’.83 The Bakaïrí data contain a possible example of a number word wearing away, as Von den Steinen notes that a word for three, ahewáo, was used ‘ebenso häufig’84 [just as often] as the longer compound form, aháge tokále. How this process works in general is important to the way we might glean information from the extinct languages of Mesopotamia, so Chapter 6 is dedicated to explaining it. Finally, what diagnostic characteristics might we expect to find in Mesopotamian numbers? Mesopotamian peoples, if they realized their numbers the same way other peoples do, would have shared the same perceptual experience of quantity we have, since it is an evolutionarily ancient system. Evidence for this would consist of the typical one-twothree-many patterning found in different kinds of number words in extant languages, as will be discussed in Chapter 6. Mesopotamian peoples would have counted on their fingers, just like we do, as this is a behavior informed by the neurological interaction between our perceptual system for quantity and our fingers. They would also have had five-fingered hands and toes, just like we possess, since this is normal for our species. Evidence would be number words showing typical finger-based constructions, as well as numerical structuring by fives, tens, and twenties. Mesopotamian peoples would have used one- and two-dimensional devices to scaffold their higher quantities and Dobrizhoffer, An Account of the Abipones, an Equestrian People of Paraguay, Vol. 2, pp. 168–169. Kroeber, Handbook of the Indians of California, p. 177. 81 Nansen, Eskimo Life, p. 194. 82 Karsten, The Head-Hunters of Western Amazonas, p. 548. 83 Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’, pp. 55–56. 84 Von den Steinen, Unter den Naturvölker Zentral-Brasiliens, p. 491. 79 80

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productive grouping, so there should be archaeological evidence of such devices and numerical evidence of such patterning. There is also a known relation between complexity in a number system and complexity in demography and the overall material culture.85 This is typically explained as a context motivating the development and application of numbers and influencing their frequency of use. This means the Mesopotamian evidence for numbers should be broadly consistent with the region’s demography and material culture: Numbers should be relatively elaborated when there are lots of people in frequent contact with one another with significant complexity to manage, relatively unelaborated for smaller groups living in more isolated circumstances with less complexity to deal with. But beyond a broad demographic/material consistency and undetailed explanations of motivation and use, there is the question of what material structures actually do in number concepts, and in our cognition more generally. I provided an initial answer at the start of this section, and as the Mesopotamian evidence for these diagnostic characteristics is reviewed in the next several chapters, I dig even deeper into this.

85 Divale, ‘Climatic Instability, Food Storage, and the Development of Numerical Counting: A Cross-Cultural Study’; Epps, ‘Growing a Numeral System: The Historical Development of Numerals in an Amazonian Language Family’.

CHAPTER 6. LANGUAGE IN HOLISTIC CONTEXT The idea that language is essential to numbers is not new. It is found, for example, in the work of philosopher John Locke, who in 1690 wrote that their names let us count and manipulate numbers, as well as conceive new ones.1 Certainly, language is a primary vehicle for storing cultural knowledge and transmitting it between individuals. This requires some context: Language stores and transmits number knowledge on the condition there are number words to label number concepts. And given this condition, language still doesn’t do this all on its own; rather, it facilitates and reinforces the material, behavioral, and social structures and processes that together comprise a culture. The question is really whether numbers have a different relation with language, relative to other human conceptual domains, and if so, how might this relation differ. Chomsky positioned numbers as a subset of language that remains inactive in some societies for reasons unspecified. There are some language-like aspects to numbers, or rather, to mathematics. At higher-order levels of syntactic representation, both mathematics and language use fixed sets of rules governing whether statements are well formed and transferring properties like truth between statements. Some context is needed here as well: Numbers are part of a system with higher-order similarities to language on the condition the numerical system has become elaborated well beyond a simple counting sequence. But Chomsky’s claim is not that an unidentified mechanism activates the number-generating capacity within language at some point, since this would involve an external switch and he envisions language as an isolated module. Rather, he identifies numbers with language so thoroughly that the language capacity either spontaneously generates numbers or somehow fails to do so. Beyond the relation between numbers and language, there is the question of the role of language in numerical origins. Some scholars ascribe to language an originating role I just don’t see when I consider the inherently alinguistic nature of both our quantity perception and advanced numerical thought (Chapter 4) or the ethnographic data of alinguistic behaviors with material forms (Chapter 5). Linguist James Hurford claimed in 1987, ‘Without language, no numeracy’.2 While our perceptual system for quantity is evolutionarily older than our capacity for language, we likely didn’t develop 1 2

Locke, An Essay Concerning Human Understanding, Book 2, Chapter XIV, Sect. 6. Hurford, Language and Number: The Emergence of a Cognitive System, p. 305.

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numbers until after we had language, so my disagreement with Hurford’s claim starts with the observation that there is little human activity language isn’t a central part of. 3 Granting what is obviously true, that language helps us access, use, and elaborate concepts like numbers, it’s still not clear how—or whether—numbers might particularly depend on language, relative to our other conceptual domains. But I do agree there is a difference, just one that runs in the opposite direction from what Hurford proposes. This can be illustrated with a parallel case, color perception and language. We see color, just like we see quantity, with a perceptual system that is pretty much the same for all primate species, just like the one for quantity is. Words help us point to colors and refer to them in their absence. Words do these things for numbers too. But with numbers, we can also represent quantity with our bodies or objects in the immediate environment, by either using our fingers or making marks on the ground. We don’t have a similar capacity to represent color to exemplify the one we mean. So even at the point where numbers are first emerging, they depend less on language—exactly the opposite of what Hurford argues—and rely more on material structures. Numbers have at least a partial independence from language; this will be a recurrent theme, especially in later chapters about material forms for numbers, particularly writing. Two decades later, Hurford would update his claim: ‘Without using language, we still can’t go [“higher than the subitizing range of about four”]’.4 In differentiating nonsubitizable quantities from subitizable ones, he appears to recognize that our perceptual experience of quantity informs how we develop numbers. He no longer positions language as necessary for the quantities we can see without counting, but thinks it’s still necessary for developing numbers in the undifferentiated many above the subitizing range. But given that we don’t perceive non-subitizable quantities without counting, it’s not clear how language would help us develop names for numbers that would enable us to count. Going back to color perception as a parallel case, it would be like expecting us to name colors we can’t see and use the names with everyone somehow knowing what we meant, despite their not being able to see the colors in question either. But here another clear role for material structures stands out: We can and do use material structures to apprehend, communicate, and use non-subitizable quantities, without necessarily also naming them in language, as the ethnographic data show. We do things like displaying a number of our fingers to represent the quantity we mean, or lining up baskets of food to determine there is an equal number of them, or making notches on a stick to accumulate a number of something. There is no compelling reason to discount such material assistance as either unimportant or separate from the process whereby number concepts are formed. Such behaviors with material forms give us insight into how numbers might emerge in the first place—and without postulating a mysterious emergence from language without material referents, like Renfrew’s concept of weight without the experience of it.5 Newmeyer, Language Form and Language Function, p. 1. Hurford, The Origins of Meaning: Language in the Light of Evolution, pp. 91–92. 5 Renfrew, ‘Commodification and Institution in Group-Oriented and Individualizing Societies’, p. 98. 3 4

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Hurford may be assuming that number words are what make number concepts discrete. There are several senses in which the word discrete is used. Two are distinct from each other and designating an exact quantity. Undoubtedly, words make numbers discrete in the sense of being distinct from each other: one differs from two, two from three, and so on, both in the sounds used and the meanings expressed. But discreteness is also what differentiates the meaning about three, which doesn’t designate an exact quantity, from the meaning exactly three, which does. As both can be labeled linguistically and the associated number word can take either meaning, it’s clear that linguistic labeling per se is not the mechanism yielding exactness from approximation. But here another role for material structures stands out: Placing about three in sequence with one through five on the fingers of the hand is a plausible mechanism for influencing the concept about three toward exactly three. What I am suggesting is that the content, structure, and organization of our number concepts comes not from language but from the material structures we use to represent and manipulate numbers. What language does is label whatever concept we happen to have, and such labels don’t reflect conceptual details like exactness or indicate when the content changes. Yet a third sense of the word discrete comes from linguistics. Sounds and words are the units involved in the generativity of language. This is discrete infinity, language’s ability to recombine a small, highly limited set of sounds and words to generate a very large, somewhat limited set of novel words and statements. For numbers, such recombination underlies our ability to generate names for higher numbers, as twenty-one combines one, two, and ten through addition and perhaps multiplication. Arguably, the amounts, relations, and operations used to produce new number words are not linguistic per se but reflect prior material practices. That is, ten-ness relates to our habit of using our fingers for counting, rather than reflecting an internal parameter of language in the Chomskian sense. Using a different material structure yields a different productive base, like the Yuki sticks produced a number system constructed on eight. Similarly, if material forms are put together differently—subtracted instead of added, or taken up in pairs instead of one by one—new combinations result in the associated number words. We end up with a number word for nine that means ten without one instead of five plus four, or we find productive terms shifting upward, as was noted for Polynesian numbers in Chapter 5. When material forms and behaviors demonstrably influence our number words, the premise that numbers or their characteristics originate in language becomes questionable. Even if I weren’t committed to the idea numerical cognition involves the active engagement of material forms, I would find it difficult to understand how it’s possible to look at number words that point to exemplars of quantity, name material structures like the fingers, group quantities by amounts related to material structures, conjoin perceptually salient quantities with operations like addition and subtraction that imply material manipulation, and associate lexical rules with motor movements,6 and then con6 Ullman et al., ‘A Neural Dissociation within Language: Evidence That the Mental Dictionary Is Part of Declarative Memory, and That Grammatical Rules Are Processed by the Procedural System’.

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clude materiality has nothing to do with number concepts. This is essentially the argument made by linguist Caleb Everett, who claims any conceptual insights associated with material forms are ‘fleeting’ and ‘ephemeral’ and therefore number concepts must come from number words.7 I don’t disagree that language is an effective vehicle for expressing numerical concepts; I just don’t think it’s the only way or their initial source. And rather than being fleeting or ephemeral, material forms persist to anchor and stabilize concepts of their quantity, like the stripes on the arm of the Wathaurung messenger.8 It’s spoken words and gestures that are the more fleeting and ephemeral of the two, a main difference between material and linguistic signs.9 Assuming language is primary may be deeply entrenched and accordingly comfortable, but it has considerable blind spots, the old effect that to a hammer, everything looks just like a nail. Reinforcing the assumption is linguists’ preference for investigating numerical language through synchronic variation, something that excludes consideration of diachronic change. This is understandable to some extent, since the synchronic view has much to offer, like defining how numerical language varies, something that can illuminate its range or potential for variability as a cognitive system. The synchronic view is also easier in many ways: Its data are more likely to have been collected under the latest linguistic methods and theories, not gleaned or reconstructed from outdated dictionaries and comparative analyses. But the synchronic view also tends to conceive numbers as monolithic entities, since change over time is the diachronic view. When the Mundurukú fuzzy three, the Oksapmin ordinal three, and the Western relational entity three are only ever considered at a time, the question of how one might be elaborated as another over time isn’t even asked, let alone considered. Numbers aren’t seen acquiring properties like linearity and manipulability, and language isn’t seen acquiring properties like productive grouping from material devices, so why these things happen and how these processes work aren’t investigated. The net result is that language remains the isolated module Chomsky proposed it to be, cognition is kept firmly sequestered in the head, and materiality is excluded as an influence on language despite the recognition that the perceptual experience of its quantity has something to do with linguistic developments. My recommendation of course isn’t to stop investigating synchronic variability in language, but rather, to approach the cognitive system for numbers holistically. In this view, language is part of a larger cognitive system, one with a manuovisual means of accessing our numerical intuitions that complements and interacts with, but is ultimately distinct from, our linguistic means. In this view, language influences and is influenced by the other components, the behaviors, material forms, and psychological capacities involved in numerical conceptualization. Interaction and influence bring in the diachronic dimension, giving us potential insight into where numbers come from and how and why they change over time. And once the emergence of a numerical lexicon Everett, Numbers and the Making of Us: Counting and the Course of Human Cultures, p. 193. Morgan, The Life and Adventures of William Buckley: Thirty-Two Years a Wanderer amongst the Aborigines of the Then Unexplored Country around Port Phillip, Now the Province of Victoria, pp. 48–49. 9 Malafouris, How Things Shape the Mind: A Theory of Material Engagement. 7 8

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‘merely reflects the development of more efficient, extralinguistic techniques’,10 language need no longer carry the whole explanatory burden for numbers. Admitting materiality into the system also helps us escape the problem with ascribing numerical origins to language: A means of invention should not presuppose that which it invents. Simply, numbers don’t emerge fully formed, like Athena from the head of Zeus. They emerge from our experience of quantity in material forms, and they elaborate through our use of material forms. The originating sequence is this: We perceive quantity with a perceptual system so evolutionarily ancient, it precedes the emergence of language both phylogenetically in our species and ontogenetically in our infants and underlies the expression of numbers in any form. Numbers emerge through behaviors with material forms, first in the subitizable range we can see and then beyond it. Linguistic labels emerge throughout but secondarily, on the condition there is enough repetition of the behaviors with material forms to occasion naming. Both number concepts and number words become encoded as material devices, behaviors, norms and habits, expectations, and words, enabling their subsequent transmission to new individuals through the enculturation process. The observation that linguistic labels not only follow but are influenced and informed by the materials we use for numbers makes linguistic data directly pertinent to an archaeological inquiry into numerical origins. Using the fingers for counting leaves characteristic traces in language in the form of words that mean both fingers and numbers, compounds for numbers like six that mean five plus one, and productive cycles of five, ten, and twenty. Thus, where we find such characteristics in archaic languages, we have evidence of ancient finger-counting. Similarly, where we find the one-two-threemany patterning characteristic of the number sense in archaic number words, we have evidence that the number sense was present in ancient populations and that it worked the same way for them as it does for us today. This positions language as a potential source of information on the material forms once used for numbers, as well as the psychological capacities of ancient peoples. This insight is particularly valuable for recovering finger-counting, which tends not to leave any physical trace behind for archaeologists to discover, and the number sense, which cannot be observed directly in extinct populations.

LEXICAL NUMBERS Lexical numbers are the words we use for counting. Some are compounds that clearly show their etymological roots in material structures like fingers or arithmetical processes like addition. Others are unanalyzable syllables revealing nothing of their antecedents. Understanding why number words differ in this regard can reveal not just the material roots of our numbers but something about their relative age. The general pattern to their emergence, as suggested by the ethnographic and linguistic data, is this: Subitizable number concepts and the associated number words precede any and all non-subitizable ones. Small non-subitizable number concepts and number words emerge next, usually through finger-counting or as simple material 10

Damerow, Abstraction and Representation: Essays on the Cultural Evolution of Thinking, p. 212.

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combinations of addition or subtraction. Emerging last are large non-subitizable number concepts and number words, those higher than what can easily be represented on the hands and feet of a single individual. Within this broad pattern—subitizable numbers, small non-subitizable numbers, and large non-subitizable numbers—each category can be specified further. In the range of subitizable numbers, words for one and two emerge first, with the chances for either to precede the other appearing about equal.11 One and two suggest both the perceptual salience of small quantities and relational judgments, the similarity of perceived quantity shared between two single objects or two pairs of objects and the dissimilarity of quantity that a single object has relative to a pair of objects. Three emerges next, not always as a term meaning about three but as the combination of two and one and occasionally one and two. This means that despite the perceptual salience of three-ness, we still put ones and twos together—implying we manipulate material forms— to explicate a concept of three. This is difficult to explain if language simply generates number words, especially for quantities that are subitizable. Of the small non-subitizable numbers, five and ten emerge first through the use of the hands to express quantity manuovisually. This is quickly followed by a name or phrase that points to the fingers or hands—as many as the fingers on one (or two) hand(s). What I call the gap numbers emerge next. These are remaining numbers in the sequence one through ten: four and six through nine. Four is found as combinations of two and two, three and one, five minus one, or a term that associates it with a specific finger or a material form like the toes of the Abipón emu. Six through nine are often, but not exclusively, compounded as five-plus the requisite smaller number, formations implying the use of the fingers. Alternatively, six is compounded as two and two and two, as in Bakaïrí; three and three, as in Waimiri, a Brazilian language; or four and two, as in Mianmin, a Papua New Guinean language.12 The numbers used to adjust a large productive number or step like five are typically subitizable. As four is the extreme upper limit of the subitizing range, back-counting13 allows numbers like nine to be realized by subtracting a subitizable number from the higher quantity, so that nine might be one from ten rather than five plus four. The preference for subitizable quantities in compounding undoubtedly reflects our ability to perceive them, but their use may also be motivated because their names are our first number words, making these labels both available and familiar for constructing new numbers. Large non-subitizable numbers tend to be higher than the numbers counted on an individual’s two hands and perhaps two feet. These numbers often emerge as multiples of the human digits, ten and twenty, steps further adjusted by counting up or occasionally down. Their names are compounds of smaller numbers put together arithmetically, so that forty-one might be four tens and one or two twenties and one, and thirty-nine might be Closs, ‘Native American Number Systems’. Rosenfelder, ‘Numbers from 1 to 10 in over 5000 Languages’; Von den Steinen, Unter den Naturvölkern Zentral-Brasiliens. Reiseschilderung und Ergebnisse der Zweiten Schingú-Expedition 1887– 1888. 13 Menninger, Number Words and Number Symbols: A Cultural History of Numbers, p. 75. 11 12

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three tens and nine or two twenties less one. Numbers that are even higher reflect the emergence of new productive terms like hundred and thousand that effectively multiply a base number like ten itself related to a material form like the fingers. More fundamentally, they reflect simple counting in a higher range: Just as counting single items from one to ten yields ten, counting groups of ten from one to ten yields a hundred. This pattern reflects more than the order in which the terms initially emerge. Subitizable and small non-subitizable number words reside in the mental lexicon, the function of our long-term memory that stores the meanings and sounds of about 10,000 words,14 and possibly as many as 80,000 ‘if all the proper names of people and places and all the idiomatic expressions are also included’.15 Storage in the mental lexicon subjects subitizable and small non-subitizable number words to memorization effects, as well as processes of linguistic change. Over time, these effects and processes reduce number words that are compounds—a category including most number words, especially if they are recent—to unanalyzable syllables. That is, instead of showing their roots in arithmetical operations or finger-counting, terms like two and one and all the fingers on one hand and two on the other eventually wear away to syllables like three and seven. Storage in the mental lexicon has the greatest effect on words for the subitizable numbers. Since subitizable number words emerge first, they have the greatest longevity of all the number words in any particular language. This gives them the greatest amount of exposure to memorization effects and processes of linguistic change. Because of this exposure and because they emerge when the number system has the least structure, subitizable number words are typically the most irregular. They are also the number words used the most frequently, something that can be seen in analyses of how languages are used. Frequency of use can be estimated by compiling a text corpus, as for example, all the words used in online English-language newspapers and periodicals within a certain range of years. These data are then used to calculate the frequency of particular words relative to all the others in the corpus. A graph of word-use frequency for the words one through twenty in English is shown in Fig. 6.1. Frequent usage increases the likelihood that words will be memorized; memorization in turn enables words to be irregular, giving subitizable number words an even greater potential for irregularity of form.16 Not surprisingly, because English is associated with a decimal number system, there are some small increases in usage around multiples of ten; there are smaller increases around multiples of five, but these are so small they are difficult to discern on a linear plot. The more a particular number word is used, the more likely it will be stored in the mental lexicon and the less likely it will have a name generated by means of a lexical rule; conversely, the less a particular number word is used, the less likely it will be stored and the more likely it will have a rule-generated name.

14 Carreiras et al., ‘Numbers Are Not like Words: Different Pathways for Literacy and Numeracy’, p. 86. 15 Aitchison, Words in the Mind: An Introduction to the Mental Lexicon, p. 8. 16 Bybee, Language, Usage and Cognition.

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In Chapter 5, I quoted several authors17 who remarked that the Yuki, Abipón, Aguaruna and Jibaros, Inuit, and Bakaïrí had only a few ‘proper’ or ‘real’ numbers. The rest of their numbers were overt compounds related to things like finger-counting. These authors may have assumed that words whose etymological basis was unanalyzable were privileged in some way, presumably because the parallel words in their own Indo-European languages were unanalyzable. They were in fact observing the result of the processes that alter number words residing in the mental lexicon. Western languages, of course, are hardly immune; once terms and phrases are in the mental lexicon, it’s merely a question of time. Given sufficient time, terms and phrases wear away to become unanalyzable syllables whose roots are no longer apparent. This has already occurred in Proto-Indo- European by the time we have our first insight into it.

Fig. 6.1. Frequency of use for the lexical numbers one through twenty in English. The subitizable numbers are the most frequently used number words, with frequency decreasing in a way that evokes the Weber–Fechner constant. There are also small increases in use frequency at multiples of ten and very small increases at multiples of five, characteristics typical of a decimal number system. Both lexicalization and grammaticalization presuppose frequency of use18; the association of lexical rules with the procedural system also implies the involvement of motor movements.19 The data shown in the graph were downloaded from the British National Corpus on 16 September 2017. As of late 2018, this text corpus was advertised as containing over 100 million words sourced from both spoken and printed media, including fictional, journalistic, and academic genres. It is available online at https://corpus.byu.edu/bnc/.

Proto-Indo-European is a language thought to have been spoken about 6500 to 4500 years ago. It has been reconstructed by comparing the vocabulary of its modern de17 Dixon and Kroeber, ‘Numeral Systems of the Languages of California’; Dobrizhoffer, An Account of the Abipones, an Equestrian People of Paraguay; Karsten, The Head-Hunters of Western Amazonas: The Life and Culture of the Jibaro Indians of Eastern Ecuador and Peru; Nansen, Eskimo Life; Von den Steinen, Unter den Naturvölkern Zentral-Brasiliens. 18 Bybee, Language, Usage and Cognition; Cacoullos and Walker, ‘Collocations in Grammaticalization and Variation’; Heine, ‘Grammaticalization’. 19 Ullman et al., ‘A Neural Dissociation within Language’.

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scendants, languages like German and English. Even at roughly 5500 years ago or about 3500 BCE, Proto-Indo-European numbers are pretty much unanalyzable: *hoi(h)nos, *duoh1, *treies, *kwetuōr, *penkwe, *(s)uéks, *séptm, *h3eḱteh3, *(h1)néun, and *déḱmt (the preceding asterisks mark their reconstructed status).20 There are, nonetheless, some lingering hints to their roots: The word five suggests the word whole, implying a phrase like the whole hand, while ten may originally have meant two of something, quite likely the hands.21 The words eleven and twelve in Proto-Germanic, a Proto-Indo-European descendent spoken in the late 1st millennium BCE, are understood to mean one left (over ten) and two left (over ten), phrases suggestive of finger-counting. While we understand why words in the mental lexicon change, we don’t really know how long it takes. There aren’t any heuristics in the linguistic literature, an outcome of prioritizing synchronic assessment over diachronic analyses. If substantial amounts of time were involved, such diachronic change would be difficult to observe in any case. While comparisons of descendent languages provide some insight into ancestral languages, these techniques can only go back so far, and the phenomenon of interest may have occurred even earlier, as appears to be the case with Proto-IndoEuropean. My guess is the linguistic change that obscures the etymological roots of number words takes longer in languages spoken by societies who don’t use numbers frequently. These tend to be small, isolated groups with little internal or external complexity to manage that would otherwise motivate the use and elaboration of numbers. Conversely, wearing-away would likely take less time for number words in languages spoken by societies using numbers frequently. Relative longer and shorter, however, are a long way away from the kind of precision we would like such estimations to have. Compounds for large non-subitizable numbers like twenty-one are not stored in the mental lexicon. This makes sense, if you consider that filling the mental lexicon with names for all the numbers there could possibly be would have little benefit, since most of the potential words for higher numbers are used very little. Simply, we don’t say a number like six hundred and thirty-three often enough for us to memorize it. And it might even be detrimental if all those number words were in the mental lexicon, not just because they would exceed its capacity, but also because they would crowd out all the other vocabulary it’s useful to have in there. Instead of lexical storage and recall, we generate names for large non-subitizable numbers whenever they are needed by means of lexical rules, a function of our language capacity. Rule-based naming or lexicalization is a critical function of language, as it enables us to generate names for numbers we don’t already know and couldn’t possibly learn or memorize. This ability is no less astonishing when we consider that we can represent and manipulate all the same numbers without language by using materially, or that its parameters come from material structures and combinations like the ten-ness of our fingers and the additivity of our one-dimensional devices. It’s also not surprising that lexical rules are neurologically associated with our procedural memory system, the long-term memory function responsible for knowing how to do things like walk, talk, and use material forms for numbers; lexi20 21

Beekes, ‘The Numerals’, p. 238. Sihler, ‘Numerals’, p. 402.

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cal rules involve the same brain regions and functions because they are ‘like skills in requiring the coordination of procedures in real time’.22 Number words generated by rules also tend to be much more regular23 than words that reside in the mental lexicon, and being generated by rules instead of residing in the mental lexicon makes words for large non-subitizable numbers relatively impervious to the effects of memory and linguistic change that alter subitizable and small non-subitizable number words over time. What would cause large non-subitizable numbers to become lexicalized? They would need to be used quite frequently, at a rate that far exceeds the need to name them in the first place. This implies two conditions: a social need for frequent large numbers and, of course, the availability and use of material devices to scaffold the requisite productive grouping and operations. How much use or time this process would require is unclear, since again, there aren’t any heuristics in the linguistic literature, the amounts of time could be substantial, and such diachronic change is difficult to observe, especially if it isn’t prioritized for investigation. While regularized words for higher numbers imply lexicalization, reconstructive analyses tend to focus on higher productive terms. Words like hundred and thousand imply the conditions are in place but don’t necessarily demonstrate that lexicalization has occurred. My guess is that lexicalization, like the linguistic change that obscures etymological roots, takes longer in languages where large numbers aren’t used frequently. Again, these would likely be societies not subject to the kinds of internal and external pressures motivating the use and elaboration of numbers. Conversely, lexicalization would likely take less time when large numbers are used with high frequency. And again, relative longer and shorter are vastly inadequate heuristics. Unanalyzable small and lexicalized high number words imply antecedents that are analyzable and not lexicalized, plus some passage of time. Unfortunately, since there are no techniques for determining how long these processes take, there is no way to determine when such antecedents might have emerged. However, there is some indication of relative age. Number systems with analyzable subitizable numbers are likely to have emerged fairly recently, especially if they occur in restricted number systems, defined in Chapter 5 as systems that counting no higher than twenty.24 Number systems with analyzable terms for six through nine are likely a bit older, given the order in which number words emerge, but they are still young compared to systems like Proto-IndoEuropean where all the small number words, both subitizable and not, have become unanalyzable. Systems where lexical rules have emerged to produce terms for higher numbers are likely the oldest. Number systems with analyzable and lexicalized numbers are not randomly distributed across the globe (Fig. 6.2). Rather, number systems with analyzable terms for subitizable numbers are concentrated where humans have lived the least amount of time, at the ends of the ancient migration arcs into Australia and South America. Number systems with analyzable terms for the finger-counting range are distributed along these migration arcs, through Papua New Guinea and western Ullman et al., ‘A Neural Dissociation within Language’, p. 267. Bybee, Language, Usage and Cognition. 24 Comrie, ‘Numeral Bases’, p. 3. 22 23

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North America. Number systems with unanalyzable small number words and lexicalized higher number words are found where humans have lived the longest, especially Africa and Eurasia. This geographic distribution does nothing more than evoke the relation between numbers and material culture, the context motivating numerical development and application and influencing numerical frequency of use mentioned briefly in Chapter 5.

Fig. 6.2. Geographic distribution of 452 languages with analyzable number words. Key: Systems with analyzable terms for three and four (triangles, in gray; n = 124); six through nine (circles, in white; n = 289); or both (plus marks, in black; n = 39; these may be difficult to discern among the signs for the first two groups but are found in Australia and South America). Interestingly, there are many systems with analyzable numbers in the fingercounting range in Africa; these could be number systems that have emerged relatively recently, or older number systems whose associated linguistic forms have not changed because of infrequent social use. Not shown: 4,390 languages with unanalyzable words for the numbers one through ten; these are distributed globally but are particularly concentrated in Africa and Eurasia. The data on number words were drawn from Rosenfelder, ‘Numbers from 1 to 10 in over 5000 Languages’, the location data were drawn from the World Atlas of Language Structures, and the graphing was performed by QGIS.com.

A final point to bear in mind is that there is no necessary forward momentum or direction to numerical elaboration. Number systems can shrink, just as they can flourish or die. What happens ultimately depends on whether and the degree to which a society invests in the requisite behaviors with material forms to sustain—or not—its use of numbers. It may also be the case that at some point numbers become so useful as a cognitive technology, so interwoven in daily life, that there is no real chance of any outcome except further investment and elaboration. To borrow a term from archaeol-

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ogist Ian Hodder, we become entangled25 in numbers; they become essential to how we think and live, to the point there’s no thinking or living without them.

GRAMMATICAL NUMBER Grammatical number is the way languages distinguish singularity from plurality. Many languages, including English, distinguish a singular and a plural, typically by adding a sound to the end of the word, as in cat and cats, or by changing an internal sound value, as in mouse and mice. This is, in effective, a one–many system. In other languages, it is possible to distinguish duals, trials, and rarely, quadrals—respectively, two, three, and four of something.26 The upper range of trials and quadrals can be fuzzy in the quantity they pick out. For many, there are languages in which it is possible to distinguish a greater amount of multiple entities from a lesser amount, or a bigger many from a smaller one. This description hopefully evokes the earlier description of our perceptual experience of quantity. Grammatical number demonstrates an underlying structure identical to that of lexical numbers when they start to emerge, and likely for the same reason: Both are structured by our perceptual experience of quantity, or more specifically, by resources like attention, object tracking, and working memory that govern our ability to subitize. Their values are not the only thing suggesting grammatical number is related to both our number sense and lexical numbers: In some languages, ‘the dual and trial forms originate from the numerals “two” and “three”, and … the [paucal] comes historically from “four”’.27 In this view, grammatical number emerges from an ‘earlier stage’ of lexical numbers, ‘independent words, which over time are likely to be reduced phonetically and semantically’.28 Occasionally mentioned as a possible source of grammatical number is person, the feature of language that distinguishes the first person I from the second person you and the third person he, she, and it. This seems unlikely, as its structure is inherently different; for example, it is questionable ‘whether we and other first person plural pronouns are really the plural of I and equivalents’.29 Grammatical number—the grammaticalization of lexical numbers—implies both the prior development of lexical numbers and a fairly robust frequency of use. Because of its relation to the Weber–Fechner constant, grammatical number also implies that the people speaking the language have the typical perceptual experience of quantity that characterizes the human species. Both implications are quite valuable when it comes to gleaning information from the extinct languages of Mesopotamia, for it opens up a window onto ancient populations that we would not otherwise have. The majority of languages have both lexical numbers and grammatical number. Fewer have lexical numbers without grammatical number. Only one has neither trait. But no language known has grammatical number without having lexical numbers, regardless of whether those lexical numbers are few or many. Interestingly, the geographHodder, Entangled: An Archaeology of the Relationships between Humans and Things. Corbett, Number. 27 Corbett, p. 21. 28 Corbett, p. 266. 29 Corbett, p. 61. 25 26

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ic distribution of these traits (Fig. 6.3) is again not random: The only language with neither trait, Pirahã, is located toward the end of the ancient migration arc into South America. Languages with only lexical numbers are distributed along these migration arcs, through Papua New Guinea and western North America. Languages with both traits are found where humans have lived the longest, especially Africa and Eurasia. This distribution is consistent with the idea that grammatical number emerges from lexical numbers. And once again, the pattern suggests the relation between numbers and material culture, something that begins to change when populations become more permanently settled, larger in size, and more in contact with their neighbors.

Fig. 6.3. Geographic distribution of 168 languages with lexical numbers and grammatical number. Key: Systems with both lexical numbers and grammatical number (circles, in white; n = 155); lexical numbers only (diamonds, in gray; n = 12); or neither trait (star, in black; n = 1, located in South America, arguably the place on the planet where humans have lived the least amount of time). This is an updated version of a study I published in Current Anthropology in 2015, ‘Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number’; all data were drawn from the World Atlas of Language Structure in 2015, and the graphing was performed by QGIS.com.

The distribution of these traits, however, is even more complex. There are languages that count to very few numbers—one, two, three, many, and maybe five and ten—that have grammatical number. Grammaticalization suggests significant frequency of use, and grammaticalized number suggests that the lexical numbers in these number systems may have been around for a substantial amount of time. Again, how much time and frequency of use the grammaticalization process requires is unclear. My previous heuristic about speed and size isn’t as useful here, since grammatical number is found in languages associated with societies that are relatively small and which have few lexical numbers, like !Xóõ, spoken in Botswana. This is quite at odds with the idea that grammaticalization entails substantial use: Why would lexical numbers remain so few with a high frequency of use? Or, might this combination indicate a number system once robust enough to grammaticalize its lexical numbers, but subsequently shrinking?

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There are also languages without grammatical number associated with large, complex societies and a high frequency of use for numbers; Cantonese is a good example of this. These pose the question of why, despite high frequency of use, lexical numbers don’t grammaticalize. Grammatical number can be modulated by animacy. That is, when a language has grammatical number, it might not be applied to everything that could be spoken of in terms of singularity and plurality. When animacy affects how grammatical number is used, it marks what is animate: beings that are alive, like people and valuable animals, or that are considered to be alive, like deities. Interestingly, the animacy scale for grammatical number runs in opposite direction to the one occasionally found in lexical numbers, where the more animate something is, the less likely it is to be counted. This typically occurs where there are lexical numbers available for counting along with cultural proscriptions against counting living beings, with the latter often taking the form of a belief that counting them would cause them harm in some way. There’s a certain logic in this if you think about the order of emergence. When number words emerge, they might not initially be applied to animate beings, since these are known as individuals. This would perhaps focus animacy in lexical numbers toward the inanimate portion of the hierarchy. Numbers grammaticalize as a function of linguistic interaction, something that takes place between animate beings, concentrating it on the animate portion of the hierarchy. Also of note is the fact that languages have different ranges of grammatical number. Some are one-many systems, like English; others are one-two-many systems, and so on. When animacy is used, languages also apply it differently to grammatical number. Some languages might include animals in the animacy hierarchy; others might not. Such differences suggest that languages were probably not in much contact with one another while grammatical number was developing, an inference I will shortly apply to the extinct languages of Mesopotamia.

ORDINAL NUMBERS Ordinal numbers are number words that mark order, like first, second, and third. The lowest two tend to originate as terms meaning something along the lines of it arrives before any of the others and it arrives next, which is why there is often such a disconnect between the words for the numbers one and two and their ordinal counterparts, first and second. Before the end of the subitizing range, however, there’s typically a much closer association, seen in the resemblance of third and three, where the lexical number word is modified to make it ordinal. In English, we do this by adding the sound -th to the end of the word. Thus, four becomes fourth, five fifth, six sixth, and so on. The astute reader will notice that adding -th to the end of a lexical number word to make it ordinal looks very much like a lexical rule, and indeed, so it is. Since higher ordinal number words are not used as much as the lower ones, something the word-frequency data clearly show, we tend not to memorize them. Instead, they are generated by means of a rule whenever they are needed.

6. LANGUAGE IN HOLISTIC CONTEXT Ordinal Number first second third fourth fifth sixth seventh eighth

English 463,566 103,621 67,037 23,227 13,150 8170 6517 5814

Word Frequency Mandarin 593.92 350.36 145.73 63.88 36.93 23.96 30.94 24.95

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Arabic 143,626 28,511 34,276 16,914 11,472 7968 6265 5996

Table 6.1. Ordinal frequency in English, Mandarin, and Arabic. Though the scales used in the three corpora differ, the word frequency for the ordinal numbers first through eighth show a similar order and distribution, generally decreasing in frequency from first through eighth and significantly decreasing around fourth. Data compiled from Davies and Gardner, A Frequency Dictionary of Contemporary American English: Word Sketches, Collocates and Thematic Lists; Kilgarriff et al., ‘Corpus-Based Vocabulary Lists for Language Learners for Nine Languages’; Xiao and McEnery, ‘The Lancaster Corpus of Mandarin Chinese’.

Word-frequency data for the ordinal numbers first through eight (Table 6.1 and Fig. 6.4) show that we use the subitizable ordinals—first, second, and third—the most, with a sharp decrease in the frequency of use around the term fourth. This effect is illustrated with data on ordinals in English, Mandarin, and Arabic. I chose these three languages for two reasons. First, they come from different language families, which makes them relatively unrelated to one another. This in turn supports the claim that ordinal frequency is cross-linguistic. Second, all these languages have large, online, scholarly text corpora from which the requisite data are freely available. The higher use of subitizable ordinals and the negligible use of non-subitizable ones will hopefully once again evoke the operation of the number sense, the way our quantity perception works. What is likely happening is that ordinal frequency of use is influenced by our ability to perceive quantity and the resources of attention, object tracking, and working memory that govern the number sense. A follow-on observation is that the relatively infrequent use of the non-subitizable ordinals, fourth and higher, can occur in conjunction with the associated lexical numbers being infinite in scope and with ordinal terms like thousandth, millionth, and billionth being both well-known and readily available. This means ordinals ranging from first to third cannot be used to argue for a limited or restricted range of lexical numbers, something that will directly apply to how we understand the linguistic evidence of Mesopotamia.

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Fig. 6.4. Ordinal frequency. The three languages being compared are English, Mandarin, and Arabic. The Weber–Fechner constant for the same range is also shown. After an initial steep decrease in frequency around third, use of the non-subitizable ordinal number words tends to remain low. Note that Arabic has a slightly higher frequency of word use for third than second; the significance of this is unknown. The frequency distribution for ordinal numbers in all three corpora resembles that of the Weber–Fechner constant, suggesting that ordinal frequency of use, like the frequency use of lexical numbers (Fig. 6.1), is related to the number sense or the underlying resources that govern it. While these distributions might suggest the number systems shown are restricted in their range of lexical numbers, in fact all three are associated with unrestricted numbers and highly numerate cultures. Data from Davies and Gardner, A Frequency Dictionary of Contemporary American English; Kilgarriff et al., ‘Corpus-Based Vocabulary Lists for Language Learners for Nine Languages’; Xiao and McEnery, ‘The Lancaster Corpus of Mandarin Chinese’.

WHAT TO LOOK FOR IN THE MESOPOTAMIAN LANGUAGES, AND WHY If the various Mesopotamian peoples were like other peoples in counting on five fingers and having the sense of number typical for the human species, then their lexical numbers should reflect things like five-plus formations and cycles of ten. Their grammatical number and their frequency of use for number words should also resemble the number sense, which is the pattern found in cross-linguistic analyses of modern languages. This will need to be adjusted for differences in the text corpora, since the corpus of Mesopotamian words compiled from a relative handful of textual artifacts is much smaller than corpora of modern languages compiled by data-mining millions of documents. Why not just assume Mesopotamian peoples had the same number sense, fingers, behaviors, and word use that other peoples do? We could simply assume these things would be the case from the evolutionary aspects of our cognitive and physiological traits. However, it’s also good to demonstrate what we can by means of evidence. This would still be an inferential argument: Characteristics X, Y, and Z in extant language are related to the number sense. If archaic languages have characteristics X, Y, and Z, they demonstrate the same sense of number in ancient peoples. This is not the same as observing Mesopotamian peoples count on their fingers or testing their number sense

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through psychological means, or having them tell us these things about themselves by describing or depicting it in some way. But as I noted in Chapter 5, it is the type of thing that can show the several Mesopotamian number systems did conform to the same general pattern demonstrated by people around the globe when they count and invent their numbers. If such things could be assumed, why would it even occur to us ask whether Mesopotamian peoples had the same number sense, fingers, and behaviors in counting and word use? The question must be explicitly asked and then answered because of something inherent to Assyriology, the discipline that studies all things Mesopotamian. The historical understanding of how numbers emerged in Mesopotamia was developed under a self-imposed mandate to base interpretations of cuneiform culture solely on the evidence of the available cuneiform texts. What could be known about the various Mesopotamian peoples would be just what they had written about themselves—no more, and no less. Cross-cultural comparisons that could illustrate how number systems typically work were thus excluded, except for Piaget’s ideas on ontogenetic development in children, which were applied to Mesopotamian societies as a whole. This led to conclusions that, if true, would make Mesopotamian peoples quite unusual in the way they developed their numbers. For example, Mesopotamian peoples have been assumed to have had few lexical numbers, if they had any at all. The reasoning was that when only a few, small ordinal numbers were found, it indicated a restricted system of lexical numbers. But as we have just seen, a restricted range for ordinals occurs even when a number system and the associated numerical lexicon are unrestricted and highly elaborated. Limited ordinal use therefore has more to do with how the number sense influences numerical language than it does the extent of the number system or the associated lexical vocabulary. It isn’t grounds for arguing for a restricted numerical vocabulary. The idea that the Mesopotamian peoples had few lexical numbers has also been influenced by the way lexical numbers are typically written, as numerical notations: 1, 2, 3, rather than one, two, three. But then, numerical notations don’t record phonetic values. That is, numerical notations are a semasiographic system, something widely known about numbers with implications that aren’t always appreciated. A semasiographic system is one whose notations can be read in any language with different words and syntactic choices.30 Examples include numbers and music. Thus, 2 + 2 = 4 can be read, with equal fidelity, as ‘two plus two equals four’, ‘adding the number two with itself gives you the number four’, ‘four is the sum of two plus two’, ‘zwei plus zwei ist gleich vier’ in German, ‘liăng jiā èr děngyú sì’ in Chinese, or ‘athnyn zayid athnyn yusawi arbe’ in Arabic, and these are just a few of the possibilities. This is an attribute in which numbers differ significantly from writing for non-numerical language, which tends to be glottographic, representing sound values in addition to semantic meaning. Since notations don’t specify phonetic values, the idea that the Mesopotamian peoples had few lexical words for numbers was able to take hold, despite all the archaeological evidence of Mesopotamian peoples having counted with complex material forms associated with counting into 30

Hyman, ‘Of Glyphs and Glottography’; Sampson, ‘Writing Systems’.

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the hundreds and thousands. Yet there’s a vast difference between not knowing what the lexical words were, something requiring the phonetic values the notations don’t contain, and knowing the Mesopotamian peoples must have had lexical words, something the notations and earlier technologies like tokens indicate in and of themselves. Semasiography and glottography relate to a fundamental difference in how things mean. Material forms for numbers mean quantity because they instantiate it: Three or six marks mean three or six because they are three or six. In comparison, material forms for non-numerical language signify their meaning, either by resemblance or convention: A picture of a jar means jar because it looks like one; a quartered circle means sheep because the people using the sign have agreed on its meaning. This representational difference needs a lot more discussion, and accordingly, I address it in more detail later. Now that we have reviewed what the brain does in numbers, how peoples behave and use material forms in numbers, and how numerical language works, we are ready to look at the ancient numbers of Mesopotamia. This is the subject of the next several chapters.

CHAPTER 7. ANCIENT LANGUAGES AND MESOPOTAMIAN NUMBERS In examining the number systems of Mesopotamia, my goals are ambitious. First, I want to achieve a new understanding of these ancient numbers, one not grounded in Piagetian theories about primitive and civilized modes of thinking that risks seeing some numbers as incomplete. Second, I want to get beyond a general relation between numerical elaboration and complexity in material culture to know why and how materiality yields the concepts of numbers we have today from our perceptual experience of quantity. These goals will require reconstructing a lengthy material sequence, one that will enable an analysis of how material forms interact with human psychological and physiological abilities—and each other—to elaborate our number sense and simple counting sequences as a relational system of numbers with two dimensions, accumulation and grouping. Part of that sequence, the fingers, will be examined here. Third, I want to show how the several Mesopotamian number systems conform to how peoples count and invent their numbers, starting with their language for numbers, the subject of this chapter. Finally, I want to show there were several independently originated numerical traditions in the region, and see if the linguistic evidence helps us gain any sense of their timeline. Some introduction to the region is needed. Mesopotamia is the name given to the ancient civilization that flourished between two great rivers, the Tigris and the Euphrates (Fig. 7.1). Roughly contiguous with the modern countries of Iraq, Syria, and Iran, it is justly famous as the region where agriculture, cities, and astronomy first developed. One of the world’s earliest writing systems, cuneiform, originated there, persisting for thousands of years before dying out. Mesopotamia was also home to one of the ancient world’s great mathematical traditions, contributing to the later mathematics of the Greeks and giving us the cycles of 60 that characterize our measurements of time and angles. Known for powerful kings like Hammurabi and Nebuchadnezzar, fractious city-states in constant, cruel warfare, and monuments like ziggurats, today it is a region where war and terrorism are fast erasing its priceless archaeological and cultural heritage, a loss exceeded only by its cost in human suffering. Less well known is that Mesopotamia was inhabited by humans well after the Levant, the eastern Mediterranean coast home to the modern countries of Israel, Palestine, Jordan, and Lebanon. Where the Levant has served as the gateway to Eurasia for as long as our ancestors have been leaving Africa, Mesopotamia wasn’t suitable for settlement until some 12,000 years ago, when climatic change marking the end of the Late Upper Paleolithic (30,000 to 12,000 107

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years ago) and the beginning of the Holocene (12,000 years ago) opened it up. A stage of rapid colonization followed in the period known as the Neolithic (8300–4500 BCE). This settlement was accompanied by such astonishing inventiveness and cultural elaboration, it has been called the Neolithic revolution.1 It included increases in group size and sedentism (15,000–10,000 BCE), as well as the development of animal husbandry and small-scale cereal cultivation and storage (by 7000 BCE) and cattle domestication (after 6500 BCE).2 This revolution set the conditions needed for the Mesopotamian cultures of the Chalcolithic or Copper Age (4500–3300 BCE) and Early Bronze Age (3300–2000 BCE) to emerge, to be succeeded themselves by the Babylonians and Assyrians. It’s inaccurate to talk about a Mesopotamian people, since Mesopotamia was home to a variety of ethnic populations and cultural groups over a significant span of time. It’s common to refer to specific groups within Mesopotamia as the Sumerians, Akkadians, and Elamites when speaking of earlier peoples, Babylonians and Assyrians for later ones. But the same caveat applies to them as well, since it’s unlikely that ancient peoples thought of themselves in terms of such identities. These designations are merely convenient modern labels for ancient geographic, linguistic, ethnic, and cultural affiliations. Beyond the terminological challenge, the Mesopotamian Bronze Age groups who developed writing and mathematics must be connected with the earlier populations of Neolithic Mesopotamia who used tokens to count, and they in turn must be connected with the even earlier populations of the Upper Paleolithic Levant who may have used tallies. I grant there is no unambiguous unbroken lineage. However, genetic analyses show that the Upper Paleolithic Levantine people associated with sedentism and cereal cultivation had thoroughly intermixed with populations originating in the Zagros Mountains by the Bronze Age, 3 so a genetic thread connects the various times and populations. Cultural diffusion also occurred between the Levantine and Neolithic populations, as this is how technologies like agriculture and domestication spread, and in fact ‘the spread of ideas and farming technology moved faster than the spread of people,… [since] the population structure of the Near East was maintained throughout the transition to agriculture’.4 Noting that numbers are a cognitive technology and linguistic domain known to diffuse often and rapidly between cultures and languages, I speculate that numbers accompanied the technologies known to have spread. The inferential leap I am making is not that numbers would have spread, but that they existed in the first place to do so. Mesopotamia has an unusual potential for reconstructing a sequence of material counting technologies, one that can give us insight into how and why material forms help us elaborate the concept of number. In this regard, we owe much to the Mesopotamian use of clay and constant warfare: Burning down the cities they’d attacked and sacked had the effect of baking artifacts made of clay, preserving them for later arChilde, Man Makes Himself. Robson, Mathematics in Ancient Iraq: A Social History. 3 Lazaridis et al., ‘Genomic Insights into the Origin of Farming in the Ancient Near East’. 4 Lazaridis et al., p. 424. 1 2

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chaeological discovery and textual interpretation. Despite this circumstance, several issues perplex the reconstruction of the sequence. Foremost among these is the fact that the archaeological record generally does not provide much insight into things like finger-counting or the earliest material forms used for counting, so that the depth of time at which counting emerged is chronically underestimated. Some of these shortfalls can be mitigated by using evidence of ancient languages from archaic texts. But as I mentioned in Chapter 6, interpretations of the linguistic evidence have tended to underestimate the availability of early number words, reinforcing the impression the associated number concepts emerged only recently and with a form that was neither complex nor abstract. Here I hope both to show things like finger-counting and support the argument at least one numerical lexicon was unrestricted.

Fig. 7.1. Sumer, Akkad, and Elam in the 3rd millennium BCE. Sumer lay west of the top of the Persian Gulf, which at the time reached inland to the cities of Ur and Lagaš, much further than is true today. Akkad lay further west and to the north of Sumer in what is now Syria, Elam to the east and north in modern-day Iran. Adapted from an image in the public domain.

In Mesopotamia, there is evidence of at least three independently originated number systems coming into contact and influencing one another. The evidence takes the form of numerical notations, the subject of Chapter 10, and numerical language, the subject of this one, where the insights of Chapter 6 are applied to the lexical numbers, grammatical number, and ordinal numbers of Sumerian, Akkadian, and Elamite. The idea of independent numerical traditions changing through contact corresponds well to genetic

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evidence of separate populations in the Levant and Zagros Mountains who would later merge.5 The linguistic evidence also shows these peoples counted on their fingers and had the same perceptual experience of quantity we do. It’s an exciting new window into ancient numeracy.

THE ANCIENT LANGUAGES OF MESOPOTAMIA: SUMERIAN, AKKADIAN, AND ELAMITE Besides the Sumerians, Akkadians, and Elamites, there were many peoples in Mesopotamia: the Amorites, Eblaites, Gutians, Hittites, Hurrians, Kassites, and Urartians.6 Thus, the potential for there to have been multiple indigenous number systems in contact with one another was significant. The idea that there were several indigenous number systems in Mesopotamia is important because the interaction and contrast between the systems would have increased the likelihood of their elaboration. Of all the number systems potentially represented by these various peoples, we are primarily interested in two—those of the Sumerians and Akkadians—because it is their languages that give us our best window on early numerical prehistory. I will mention three others: ternal counting, a Sumerian system7; eme-sal, possibly a Sumerian dialect spoken by women and eunuchs8 that had a few distinct numbers;9 and Elamite numbers, a decimal adaptation of the Sumerian numerical notations that suggests the Elamites too had an independent numerical tradition. The Sumerian language was spoken between the 4th and 3rd millennia BCE. By about 2000 BCE, it had effectively become extinct, though it would remain in use for another two thousand years as a classical or scholarly language, much like Latin and ancient Greek have today in academic and religious traditions. The Akkadian and Elamite languages were spoken between the 3rd and 1st millennia BCE. All three of these languages are known primarily from their association with one of the world’s earliest original inventions of writing, as well as the literate and mathematical traditions that would develop from it. As I’ll discuss in greater detail in Chapter 10, the earliest notations for numbers were impressions made in clay, and the earliest signs for nonnumerical language labeled the commodities counted by the numerical impressions. These non-numerical signs had one of two forms. Pictures conveyed their meaning through resemblance: A picture of a sheaf of grain meant grain because it looked like a sheaf of grain. Conventionalized signs were meaningful through social agreement: A quartered circle meant ungulate, a category of hoofed animals that includes sheep, goats, and cattle, because everyone agreed that’s what it meant. But pictures and conventionLazaridis et al., ‘Genomic Insights into the Origin of Farming in the Ancient Near East’. Myers, ‘Ethnicity and Language’; Woods, ‘The Earliest Mesopotamian Writing’. 7 Blažek, Numerals: Comparative Etymological Analyses and Their Implications: Saharan, Nubian, Egyptian, Berber, Kartvelian, Uralic, Altaic and Indo-European Languages; Lambert, ‘Review, O. R. Gurney and P. Hulin, The Sultantepe Tablets II’. 8 Diakonoff, ‘Ancient Writing and Ancient Written Language: Pitfalls and Peculiarities in the Study of Sumerian’, pp. 114–115. 9 Whittaker, ‘Linguistic Anthropology and the Study of Emesal as (a) Women’s Language’. 5 6

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alized signs only provide insight into an approximate semantic meaning. For the spoken forms of words, writing must also convey sound values, and it would take some time for phonography—techniques and conventions for representing the sounds of language visually—to be invented. One way to recover the spoken form of ancient words is working backward from later scripts that do incorporate phonography to earlier scripts that don’t, looking for resemblances of character form and contexts of use that can relate earlier characters to later ones. Ideally, earlier and later states of writing would involve the same language with little change. Establishing that they are the same language involves a detailed study of texts and contexts. Temporally indistinguishable states of language are unlikely to be the case, however, since languages change over time, sometimes considerably. Consider the difference between modern English and Old English, the language of Beowulf, which are separated by much less than a thousand years. Not only will the sounds have changed, the way they are conveyed, or not, by the writing system will likely have changed as well. Certainly, Mesopotamian writing underwent substantial change during its first several thousand years as it reorganized toward increasing fidelity to language, including the representation of the sounds of speech.10 Another technique for recovering ancient spoken forms involves looking at related languages. But as Sumerian, Akkadian, and Elamite are all extinct, there are no descendent languages to compare them to. Sumerian and Elamite are also linguistic isolates, languages that are unrelated to any other known language, including each other. Sumerian is the language often associated with the earliest writing, though this identification is neither unambiguous nor uncontested. A substantial portion of what is known about Sumerian comes from Akkadian transcriptions. This means ‘we see Sumerian through an Akkadian glass darkly’,11 first because Akkadian scribes were increasingly likely to have written Sumerian as a language acquired professionally, not one spoken natively, and second because Akkadian is a Semitic language whose sound values are inferred from other Semitic languages. More specifically, Akkadian is an East Semitic language, an extinct branch of the Semitic language family, so pronunciation inferences are drawn from West Semitic languages like Arabic that are related but likely differ in pronunciation, at least to some extent. So, the Akkadian window on Sumerian pronunciation is an inference seen through an inference seen through a writing system with an initially limited ability to represent the sounds of language. For its part, Elamite remains largely unknown, especially in its earliest written form, though attempts to decipher it are ongoing.12 A final complication is that Sumerian and Akkadian significantly influenced one another, not only through bilingualism and contact, but through their written forms as well: The ‘influence of Sumerian on Akkadian (and vice versa) is evident in all areas, from lexical borrowing on a massive scale, to syntactic, morphological, and phonological convergence. This has prompted scholars to refer to Sumerian and Akkadian in the Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’. Edzard, Sumerian Grammar, p. 7. 12 Englund, ‘The State of Decipherment of Proto-Elamite’. 10 11

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third millennium as a “Sprachbund”’,13 languages that develop common features because of the geographical proximity and frequent contact of their speakers. Essentially, extensive contact, widespread bilingualism, and use of a common writing system influenced Sumerian and Akkadian to become more alike. While this phenomenon complicates inferences about the earlier, pre-contact states of either language, it also implies that matters remaining distinctive despite all the borrowing and co-influence are more likely to have reflected earlier states of independently originated traditions. Fortunately for the present inquiry, numbers fall into this category.

EVIDENCE OF NUMERICAL LANGUAGE FROM ANCIENT WRITING Writing was invented in Mesopotamia in the mid-to-late 4th millennium BCE. The script that would develop from the early impressions, pictures, and conventionalized signs would ultimately become known as cuneiform, named after the characteristic wedgeshaped impressions used to form its characters. Cuneiform was adapted to multiple languages as it spread throughout the region, and it would remain in use for several thousand years, a substantial portion of which occurred while Sumerian was gradually dying out as a spoken language.14 The earliest writing, however, consisted only of numerical impressions made with a variety of implements, including the fingers. Such impressions contain no trace of the associated number words, just the implication they would have existed, since this is the condition found in all other known number systems. When small pictures and conventionalized signs were added to label the commodities being counted, a development known as proto-cuneiform,15 the associated language remains indeterminate. This is because proto-cuneiform signs, like the numerical impressions, were semantically meaningful without the phonetic component needed to specify the intended words and determine the language used. Such notations are semasiographic: They can ‘be read with similar facility by speakers of different languages, or … [their] reading has the character of paraphrase (i.e. two different “readings” are likely to employ significant differences in word choice or syntactic construction’).16 Later writing for non-numerical language would include the phonetic component, making it glottographic, or able to represent the sounds of a particular language with fidelity.17 The difference is that written numbers tend to remain semasiographic, as this form makes them much more usable as numbers, while notations for non-numerical language are under pressure to provide pronunciation clues, since they are otherwise ambiguous as to the words intended.

13 Deutscher, Syntactic Change in Akkadian: The Evolution of Sentential Complementation, pp. 20–21. 14 Geller, ‘The Last Wedge’. 15 Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. 16 Hyman, ‘Of Glyphs and Glottography’, p. 234. 17 Sampson, ‘Writing Systems’.

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The lack of phonetic information means the language associated with the earliest impressions and proto-cuneiform writing cannot be identified with certainty.18 While many scholars have thought the language was likely Sumerian,19 a few have disagreed, as the readings of certain signs suggest ‘loans from an unknown language’,20 possibly Akkadian. Complicating the assessment is the circumstance that the early writing was succeeded by several centuries in which it degraded, suggesting a period of significant social upheaval. When writing resumed, it had a lot in common with earlier writing, and the associated language is identifiable as Sumerian. Given the degree to which the writing system was shared between languages, this circumstance cannot guarantee the earliest written forms represented Sumerian. While there is no unbroken lineage between the earliest writing and Sumerian writing, there is nonetheless significant continuity in the numbers, not just terms of the signs used, but in their structure and organization. That is, if someone other than the Sumerians invented the earliest writing, including its numbers, the Sumerians would have done something fairly uncommon by adopting and restarting it, not just successfully but without significant alteration. This is not what you’d expect if the earlier users were a different cultural group. Phonetically specified lexical numbers and unambiguous ordinal numbers and grammatical number don’t appear in texts until the 3rd millennium BCE, many centuries after writing was invented.21 As was noted in Chapter 6, lexical numbers are the number words that make up a counting sequence, as in one, two, three; grammatical number distinguishes singulars and plurals, as in cat and cats; and ordinal numbers are number words that designate order, as in first, second, and third. Assyriologists have interpreted the absence of these in writing before the 3rd millennium BCE to mean archaic peoples had only a restricted lexicon for numbers, counting no higher than twenty and possibly no higher than three. However, as has been said often enough to become passé, absence of evidence is not evidence of absence. The lack of phonetically specified lexical numbers and unambiguous ordinal numbers and grammatical number prior to their appearance in 3rd-millennium texts does not entail archaic languages necessarily lacked these features, though that is one potential explanation. A plausible and frankly more likely explanation is that early writing did not convey these things because it simply lacked the ability or motivation to do so. Certainly, the notations themselves attest to the existence of at least one language with an unrestricted numerical lexicon in the 4th millennium BCE, since tokens dated to between 8500 and 3500 BCE are thought to have represented the number ‘600’,22 as will be discussed in Chapter 9. Such material representations imply an associated vocabulary of number words whose extent was comparable to that of the material forms, even if we don’t know what the words were or the lan-

Englund, ‘Texts from the Late Uruk Period’. Veldhuis, History of the Cuneiform Lexical Tradition. 20 Englund, ‘Texts from the Late Uruk Period’, p. 80. 21 Dahl, ‘Comment on “Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number”’. 22 Cuneiform Digital Library. 18 19

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guage they came from because the writing system was relatively inexpressive at the time. Complicating this issue further is the assumption that tokens were the first counting technology used in Mesopotamia. If this were true, and I contend it is not, the timeline for developing the lexical and grammatical features of number found in the 3rd millennium BCE would be severely compressed. While the amount of time required for lexicalization and grammaticalization is unknown, the geographic distribution of these features in extant languages (Chapter 6) makes it unlikely the Mesopotamian peoples could have borrowed or developed lexical numbers, numerical accounting, and complex notations—and developed grammatical number and ordinal numbers—all within a relatively few centuries. Making it even more unlikely is that something—most likely significant social upheaval—disrupted the writing system for several of the centuries in question; this would not be the optimum condition for elaborating cultural systems like writing and numbers, even at a slow pace, let alone an unusually rapid one. The appearance of lexicalized numbers and grammatical number in 3rd-millennium texts is more likely to be a function of increased expressiveness in the writing system than represent extensive linguistic change, especially change realized at an uncommonly fast rate. Phonetic representations of Sumerian number words are attested as early as the Early Dynastic period IIIb (2700–2340 BCE), hundreds of years after writing began, and well after the inclusion of phonography for other vocabulary. Why did it take so long to represent number words phonetically? One reason, as has already been mentioned, is that it takes time to develop techniques and conventions for visually representing the sounds of language, and number words are not exempt from this. Another reason, one particular to numbers, is that numerical signs are generally more usable as numbers without phonetic information. Numerical notations combine semantic meaningfulness with concision of form, and this makes them able to represent numerical, spatial, topological, and geometric relations, facilitates our ability to recognize patterns in the information they represent, and increases the usability and manipulability of the numerical information.23 In contrast, including phonetic information detracts from their ability to do these things, the difference between calculating with numerals—1, 2, 3—and calculating with words, sentences, and paragraphs. A third reason is that numerical notations are unambiguous regarding what they mean, and they mean in a way independent of whatever they happen to be called in language. This is the difference between using Roman numerals and also knowing their names in Latin. For this reason, semasiographic representation may suffice for lexical numbers long after the writing system improves its ability to represent the sounds of language, explaining why phonetic pronunciations of Sumerian lexical numbers are rarely found24 and adding to the underestimation of their availability.

23 Larkin and Simon, ‘Why a Diagram Is (Sometimes) Worth Ten Thousand Words’; Sfard and Linchevski, ‘The Gains and the Pitfalls of Reification? The Case of Algebra’. 24 Jagersma, A Descriptive Grammar of Sumerian.

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Fig. 7.2. Lexical tablet (TM.75.G.02198) with phonetic values for the Sumerian numbers two through ten. (Left, photograph, and middle, transcription) The cuneiform tablet was found in the Royal Palace G in the city of Ebla. It has been dated to 2350–2250 BCE. It was last reported held at the National Museum of Syria at Idlib. (Right, translation) The Sumerian number word given for one, dili, appears in brackets because it is not phonetically specified, and other sources were used to identify its phonetic value.25 Left and middle images originally published in Edzard, ‘Sumerisch 1 bis 10 in Ebla’, Fig. 26a-b, with photo credited to M. Necci. These images are republished with the permission of Missione Archeologica Italiana in Siria. The image on the right was created by the author and modeled on Friberg, ‘The early roots of Babylonian mathematics III. Three remarkable texts from ancient Ebla’, p. 5, Fig. 1b.

An example of phonetically represented Sumerian number words is shown in Fig. 7.2. The artifact comes from 24th-century BCE Ebla, a city whose inhabitants spoke a Semitic language known as Eblaite.26 The context in which it was found suggests the Eblaitespeaking scribe who made it likely wanted to learn the Sumerian number words along with the Sumerian number signs, much like someone today might want to learn the Latin words for numbers along with Roman numerals.

LEXICAL NUMBERS Sumerian numbers were sexagesimal, or base 60, combining a sub-base of 10 and a base of 6 to produce alternating cycles of 10 and 6.27 While base 60 is relatively uncommon among number systems, Sumerian numbers are typical in all other respects (Fig. 7.3). The smallest Sumerian numbers are the most irregular, and the numbers one through five are unanalyzable or atomic terms. This is consistent with their longevity within the number system and the processes of linguistic change discussed in Chapter 6. Numbers become regular at 60 and higher, suggesting lexicalization, or production 25 Diakonoff, ‘Some Reflections on Numerals in Sumerian: Towards a History of Mathematical Speculation’; Edzard, ‘Sumerisch 1 bis 10 in Ebla’; Friberg, ‘The Early Roots of Babylonian Mathematics III. Three Remarkable Texts from Ancient Ebla’; Pettinato, Materiali Epigrafici de Ebla – 3 (MEE 3). Testi Lessicali Monolingui Della Biblioteca L. 2769. Seminario de Studi Asiatici, Series Maior III. 26 Edzard, ‘Sumerisch 1 bis 10 in Ebla’. 27 Thureau-Dangin, ‘Sketch of a History of the Sexagesimal System’, p. 104.

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by linguistic rules. Between these two ranges, numbers are irregular identifiable compounds, most likely produced by counting with the fingers for the numbers five through ten and with the toes for numbers up to twenty. Unanalyzable productive terms occur where they are predicted to occur, at regular multiples of the base: 60 (601), 3600 (602), 216,000 (603), and 12,96,000 (604). What the Sumerian lexical numbers tell us is that the people who invented them counted on their fingers and toes, like other peoples do: ‘Where terms for 5 to 10 exist, these are usually based on “hand”, and terms for 10 to 20 on “foot”; crosslinguistically, these are typical sources for these values’.28 In fact, up to sixty, the point at which Sumerian numbers become unambiguously sexagesimal, they would be classified as either a quinary–decimal or quinary–vigesimal system: quinary because the majority of the expressions six through nine are formed as 5+x, where x ranges 1–4; decimal because expressions like thirty are formed as 10x, where x ranges 1–9; and vigesimal because expressions like forty are formed as 20x, where x ranges 1–9.29 Mathematician Abraham Seidenberg combined them, writing ‘the Sumerian number system could be called a quinary–vigesimal system, with traces of decimal counting’.30 The Sumerians likely named their higher numbers for the material forms used to represent them, as šar (602) meant ‘ball’ and šar-gal (603) meant ‘big ball’, terms evoking the Neolithic tokens and 4th-millennium BCE impressions.31 This is consistent with the idea that material forms of expressing numbers precede and occasion lexical naming. Sumerian lexical numbers also appear to have changed in ways typical to number systems generally, with processes of linguistic change yielding the unanalyzability of the terms for one through five. The overt compounding in the numbers six through nine suggest the Sumerian number system was somewhat younger than that of the Akkadians, whose numbers in this range are unanalyzable.

Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’, p. 67. Suggested by the classification criteria in Schapper and Hammarström, ‘Innovative Numerals in Malayo-Polynesian Languages Outside of Oceania’, p. 424. 30 Seidenberg, ‘The Sixty System of Sumer’, p. 440. 31 Powell, ‘Maße und Gewichte’, pp. 480–481. 28 29

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Fig. 7.3. Sumerian lexical numbers. Productive multiples of 60 are highlighted in gray and bold; productive multiples of 10 appear in bold. Phonetic values for the numbers marked with asterisks were unknown when Powell conducted his analysis. He assumed these lexically unspecified numbers were read as they were written, based on their omission from the lexical lists, glossaries whose purported goal was an otherwise exhaustive compendium of Sumerian signs.32 Although the word for eight is not an overt five-plus compound,33 the words for six, seven, and nine are, suggesting finger-counting. Similarly, compounds of ten and twenty in numbers up to sixty suggest the use of the hands and feet for counting. Higher numbers were likely named after the material forms used to represent them, which were ‘counters (tokens), perhaps šar, “ball”, šargal, “big ball”’.34 While Powell thought the Sumerian nomenclature for the higher orders had ‘no trace of any systematic nomenclature which might reflect mathematical speculation and systematization’,35 the Sumerian higher orders emerge regularly at the predicted places—exponents of 60 and multiples of 10—so their naming is no less systematic than that of other languages. Compiled from Powell, Sumerian Numeration and Metrology. Powell, Sumerian Numeration and Metrology, p. 46. Edzard, ‘Sumerisch 1 bis 10 in Ebla’. 34 Powell, ‘Maße und Gewichte’, pp. 480–481. 35 Powell, Sumerian Numeration and Metrology, p. 78. 32 33

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Sumerian ternal counting (Table 7.1) is found in scribal and priestly contexts,36 which tended to be male in their demographic composition. Ternal numbers differ radically from Sumerian lexical numbers in both organization and vocabulary. These numbers appear to have been restricted: Not only are they few, their method of compounding wouldn’t produce higher quantities efficiently. They nicely illustrate the principle that selecting a small number like three as the productive base creates names for higher numbers that are relatively long and cumbersome: With the logic used in ternal naming and without higher productive terms, the term for nine would be PEŠ-PEŠ-PEŠ, the term for twelve PEŠ-PEŠ-PEŠ-PEŠ, and the term for sixty an unwieldy 20 PEŠ, or PEŠ-PEŠ-PEŠ-PEŠ-PEŠ-PEŠ-PEŠ-PEŠ -PEŠ -PEŠ-PEŠ -PEŠ -PEŠ -PEŠ -PEŠ -PEŠPEŠ-PEŠ-PEŠ-PEŠ. Similar compounding is found in extant number systems; these tend to be restricted, associated with traditional societies, and classified as emerging. Bakaïrí numbers, discussed in Chapter 6, are an example: One is tokále; two is aháge; three is aháge tokále [or one and two]; four is aháge aháge [or two and two]; five is aháge aháge tokále [or two and two and one]; and six is aháge aháge aháge [or two and two and two].37 Another is found in the Juwal language of Papua New Guinea, where the word for forty is the word for two, nevinja, repeated twenty times,38 though this might not be a word per se but counting represented iconically in sound. These similarities suggest ternal numbers may have represented the retention of an older numerical tradition or the inclusion of a minority numerical tradition. Nr. 1 2 3 4 5 6 7 Nr. 1 2 3 60

Ternal Counting ge daḫ PEŠ PEŠ-ge or PEŠ-bala PEŠ-bala-gi4 PEŠ-bala-gi4-gi4 PEŠ-PEŠ-gi4 Eme-sal de nim am(m)uš mu-uš, min-eš

Gloss 1 addition next? 3 + 1 or 3 passed 4+1 4+1+1 3+3+1 Gloss Not enough information to analyze

Table 7.1. Sumerian ternal counting (left) and eme-sal numbers (right). Both of these may have reflected older or minority numerical traditions, though eme-sal is generally understood as a Sumerian dialect likely spoken by women and eunuchs. Data from Blažek, Numerals: Comparative Etymological Analyses and Their Implications: Saharan, Nubian, Egyptian, Berber, Kartvelian, Uralic, Altaic and Indo-European Languages, p. 329.

Lambert, ‘Review, O. R. Gurney and P. Hulin, The Sultantepe Tablets II’. Von den Steinen, Unter den Naturvölkern Zentral-Brasiliens. Reiseschilderung und Ergebnisse der Zweiten Schingú-Expedition 1887–1888, p. 406. 38 Saras, ‘Juwal, Papua New Guinea’. 36 37

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Eme-sal is thought to have been a Sumerian dialect spoken by women and eunuchs.39 It includes a few numbers,40 and if these are construed to somewhat resemble Sumerian lexical numbers, it is only on the basis of things like reversing the Sumerian word for two, min, as nim. Since eme-sal is found in literary contexts, it was perhaps a literary convention rather than a spoken dialect,41 though numbers would be a curiously specific detail for an invented convention. Further, aspects to Sumerian eme-sal numbers are consistent with how such women’s dialects are understood generally. The eme-sal form for 60, min-eš, suggests the reading two thirties, as min was two and êš three in the Standard and Eblaite dialects of Sumerian.42 This overt compounding stands in contrast to the unanalyzable Sumerian form, ĝeš, and evokes one of three evolutionary paths for developing a women’s dialect, gendered speech changing at different rates, which characterizes five of 14 extant languages with women’s dialects.43 Though Assyriologist Igor Diakonoff (1975) found ‘no indications that [eme-sal was] a territorial or a tribal dialect’,44 nonetheless, it too possibly reflected the remnants of an older or a minority tradition. The cross-cultural tendency to take women as wives, concubines, servants, and slaves characterized the Ancient Near East as well, and the resultant ‘dialect merger’ is another of the three evolutionary paths to developing a woman’s dialect.45 The possible if tenuous resemblances between eme-sal and Sumerian numbers are not dissimilar to pronunciation distinctions in extant languages: Men and women pronounce the word two differently in Chukchi,46 one of two extant languages whose women’s dialect lists ‘dialect merger’ as its evolutionary path.47 Eme-sal numbers are really too few, however, to draw any firm conclusions.

39 Whittaker, ‘Linguistic Anthropology and the Study of Emesal as (a) Women’s Language’. 40 Blažek, Numerals, p. 329. 41 Whittaker. 42 Blažek, Numerals, p. 329. 43 Dunn, ‘Gender Determined Dialect Variation’, p. 46. 44 Diakonoff, ‘Ancient Writing and Ancient Written Language’, p. 113. 45 Dunn, ‘Gender Determined Dialect Variation’, p. 46. 46 Dunn, ‘Chukchi Numerals’. 47 Dunn, ‘Gender Determined Dialect Variation’, p. 48.

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THE MATERIAL ORIGIN OF NUMBERS Nr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Word ištēn šinā šalāšat erbet(ti) ḫamšat šeššet sebet(ti) samānat tišīt eš(e)ret ištēššeret šinšeret šalāššeret erbēšeret ḫamiššeret šeššeret

Gloss 1 2 3 4 5 6 7 8 9 10 1 + 10 2 + 10 3 + 10 4 + 10 5 + 10 6 + 10

Nr. 17 18 19 20 30 40 50 60 90 100 120 600 1000 3600 10,000 216,000

Word sebēšeret samāššeret tišēšeret ešrā šalāšā erbeā, erbâ ḫamšā šūš(i), šūsum, šu tišeā meat, meatum šinā šuši nēr, nērum līm(i), līmum šār, šārum nubi, ra/ibbatu šuššar

Gloss 7 + 10 8 + 10 9 + 10 10 × 2 3 × 10 4 × 10 5 × 10 6 × 10 9 × 10 100 120 600 borrowed? Sumerian 10,000 60 šar

Table 7.2. Akkadian lexical numbers. Akkadian numbers are clearly a decimal system, as Semitic languages tend to be. They show a Sumerian influence in borrowing number words like šār (3600). Only the common masculine form is shown. The feminine and alternative forms are omitted for the sake of simplicity; for these, consult the original material. Data compiled from Miller and Shipp, An Akkadian Handbook: Helps, Paradigms, Glossary, Logograms, and Sign List; Powell, ‘Notes on Akkadian Numbers and Number Syntax’.

In the 2nd and 1st millennia BCE, the ‘predominantly Semitic population … normally counted with decimal numbers, using their own Semitic decimal number words’,48 shown in Table 7.2. Known for borrowing Sumerian writing, the Semitic Eblaites ‘knew all the cuneiform signs expressing numbers, but they still prefer a system with a decimal base. This appears most clearly in the numbers above 100 which are always expressed by this system and not by the sexagesimal prevalent in Mesopotamia’.49 Decimal organization not only supports the idea of an independent origins for Akkadian numbers, it implies these numbers likely developed through finger-counting, reinforcing the commonality of their invention process. The idea that Akkadian lexical numbers originated independently of Sumerian numbers is also supported by the lack of any similarity between the spoken forms of the lower Akkadian numbers and their Sumerian counterparts. Higher Akkadian numbers were clearly influenced by contact with Sumerian numbers, shown by Akkadian borrowing number words like šār (3600) and Sumerian numerical notations. The Akkadians may also have been more analytical of numerical structure, since the Akkadian word for 603 was not taken verbatim from Sumerian, where it was šar-gal or ‘big šar’, but was instead called šuššar, ‘sixty šar’.50 In Friberg, A Remarkable Collection of Babylonian Mathematical Texts, p. 5. Pettinato, The Archives of Ebla: An Empire Inscribed in Clay, p. 183. 50 Powell, Sumerian Numeration and Metrology, p. 78. 48 49

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addition, the unanalyzability of the numbers one through ten suggest that the Akkadian number system was somewhat older than that of the Sumerians, whose numbers six through nine were still analyzable compounds. As for the Elamite language, its numbers are currently known only as semasiographic notations, which do not allow their spoken form to be recovered. Here I will merely note that their numbers too were decimal,51 suggesting finger-counting and an origin independent of Sumerian numbers.

GRAMMATICAL NUMBER Grammatical number is a feature of language that points to the quantity of items referred to in speech, a function known as deictic encoding.52 Across languages, this deictic encoding is structured in a way that reflects our perceptual experience of quantity: one-two-three and occasionally four, and many, subdivided as big many and small many.53 As a comparison, English is a one-many system. There are eight ways to mark number,54 and individual languages may employ several. English uses plural suffixing, adding -s, -es, and -ies to the end of nouns to mark their plurality: cat (single, meaning one; unmarked form) and cats (plural, meaning many; form marked by the suffix -s). English also uses a few internal vowel changes, as in man/men; retains formations like ox/oxen from Old English; and adopts foreign conventions like criterion/criteria. Grammatical number, when it is present, might be modulated by animacy; where this is the case, nouns for animate beings will be marked, while nouns for inanimate objects will not. English is insensitive to animacy: It is obligatory to mark number for all nouns, regardless of their animacy status. In sum, grammatical number has three aspects: its structure, within the general limits imposed by the perceptual experience of quantity or the cognitive resources governing it; its method(s) of linguistically marking the various gradations of structure used; and whether and how animacy modulates its expression. Differences in these aspects imply grammatical number developed independently in the languages being compared. All three of the ancient languages we are interested in here had grammatical number but differed in the details of their structure, marking method(s), and animacy (Table 7.3). Sumerian had a one-many structure modulated by animacy, distinguishing singulars and plurals for animate nouns but not inanimate objects through suffixing, reduplication, and encliticization.55 Reduplication repeats a word to indicate its plurality: An example in English would be mountain-mountain. Encliticization is the use of a phrase-final clitic like the Sumerian plural marker {ene}; a clitic is a morpheme that Englund, ‘Proto-Elamite’; Englund, ‘The State of Decipherment of Proto-Elamite’. Corbett, Number; Perkins, Deixis, Grammar, and Culture. 53 Franzon, Zanini, and Rugani, ‘Do Non-Verbal Number Systems Shape Grammar? Numerical Cognition and Number Morphology Compared’; Overmann, ‘Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number’. 54 Dryer, ‘Coding of Nominal Plurality’. 55 Michalowski, ‘Sumerian’; Rubio, ‘The Languages of the Ancient Near East’; Edzard, Sumerian Grammar. 51 52

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acts like a word syntactically but cannot exist on its own.56 Akkadian had a one-two-many structure because of an additional dual form used with a small number of nouns; it marked number with plural suffixing and was insensitive to animacy.57 Like Sumerian, Elamite had a one-many structure modulated by animacy, but it marked grammatical number with plural suffixing and reduplication only.58 Language Structure Sumerian One-many

Animacy Yes, animate nouns only

Akkadian Elamite

No Yes, animate nouns only No

English

Method(s) of Marking Plural suffixing, reduplication, and encliticization One-two-many Plural suffixing One-many Plural suffixing and reduplication One-many Plural suffixing

Table 7.3. Grammatical number in Sumerian, Akkadian, Elamite, and English. Differences suggest grammatical number developed independently in the three ancient languages. Data compiled from multiple sources.

The differences in these attributes suggest that grammatical number developed independently in Sumerian, Akkadian, and Elamite. Independent development would have occurred before the languages came into significant contact with one another. This means grammatical number would have developed before the Neolithic period of cultural contact and co-influence. This in turn means that numbers emerged in the region much earlier than is generally accepted. Grammatical number has two further implications. First, as the grammaticalization of lexical numbers, it affirms their prior existence. While this eliminates any possibility that Sumerian, Akkadian, and Elamite lacked number words altogether, it does not attest to anything more than a restricted numerical lexicon for any of these languages (see Chapter 6). Arguments for their having unrestricted numerical lexicons must therefore be made on other grounds, like the use of material structures for representing numbers and its implication for an associated lexicon or the general relation between the complexity of material culture and the complexity of its number system. But it does mean ancient peoples likely had at least restricted numbers by the onset of the Neolithic. Second, grammatical number conforms to our perceptual experience of quantity. In extant languages, this linguistic structuring represents subitizing or the resources of attention, memory, and object tracking that govern subitization, as well as the ability to appreciate quantity differentials above the subitizing range. In extinct languages like those of Mesopotamia, this linguistic strucJagersma, A Descriptive Grammar of Sumerian. Edzard, Sumerian Grammar; Goetze, ‘The Akkadian Masculine Plural in -Ānū/ī and Its Semitic Background’; Huehnergard and Woods, ‘Akkadian and Eblaite’; Sayce, An Elementary Grammar; with Full Syllabary and Progressive Reading Book, of the Assyrian Language, in the Cuneiform Type. 58 Grillot-Susini, Éléments de Grammaire Élamite; Khačikjan, The Elamite Language; Stolper, ‘Elamite’. 56 57

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turing shows the presence and similar operation of the same cognitive processes. In short, the fact that their grammatical number conforms to the same limits as ours shows that ancient Mesopotamian peoples had the same perceptual experience of quantity we do. While this is only to be expected, given the phylogenetic distribution of the number sense, it’s nonetheless gratifying to be able to confirm that it is so through an analysis of the linguistic evidence.

ORDINAL NUMBERS Ordinal numbers express order numerically. Order can also be expressed nonnumerically, as in before, next, and following. Across languages, non-numerical terms for order account for differences between the lowest lexical and ordinal numbers: one and first; two and second. This, in turn, suggests that non-numerical conventions either emerged before lexical numbers did or were in use before lexical numbers were adopted to express order numerically. Once numbers are adopted to express order, there is a high similarity between the lexical and ordinal form: three and third; four and fourth, and so on. Since three and sometimes four are subitizable, they suggest ordinals can develop in conjunction with restricted numbers. However, in a sample of 113 extant languages from the World Atlas of Language Structures database, none with restricted numbers had ordinal numbers.59 This finding suggests that lexical numbers may become grammaticalized as ordinals only after they become unrestricted. On this interpretation, the presence of ordinal numbers suggests an unrestricted numerical lexicon. Left unanswered is the question of when ordinals emerged in Sumerian, which would help pinpoint when the associated numerical lexicon could be classified as unrestricted. While there is currently no way to determine the time required for such lexicalization to develop, the appearance of ordinals in 3rd-millennium texts is more likely a result of the writing system becoming better able to express existing features of language than it was to have reflected their initial development. Unfortunately, the linguistic evidence brings us no closer to answering the question of when the Sumerians may have developed unrestricted numbers. For this, we will need to examine the archaeological evidence. In writing, ordinals are easily represented by combining semasiographic number signs with a phonographic sign for a sound value. In Mandarin, this convention takes the form of the prefix ➨ added to the numeral: ➨୍ for first, ➨஧ for second, ➨୕ for third, and so on.60 In English, this convention takes the form of a suffix added to the numeral: 1st, 2nd, 3rd, 4th, and so on. Sumerian ordinals were formed similarly by adding a suffix representing a sound in the Sumerian language to a number sign: {kamma} was used in the mid-3rd millennium BCE, {kam} in the late-3rd millennium.61 The resultant ordinal form was 1(diš)-kam for first; 2(diš)-kam for second; and so on. The fact that the Sumerians used the same method found in extant languages supports the claim that Sumerian numbers and numerical language worked just like extant ones. It also 59 Overmann, ‘Numerosity Structures the Expression of Quantity in Lexical Numbers and Grammatical Number’, supplemental information. 60 Xiao and McEnery, ‘The Lancaster Corpus of Mandarin Chinese’. 61 Jagersma, A Descriptive Grammar of Sumerian.

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suggests that the numerical component of Sumerian ordinals similarly represented not just the number signs but the associated word for the number as well, demonstrating the availability of a numerical lexicon comparable to the material forms used to represent numbers. Ordinal numbers are subject to frequency effects in spoken and written language, with subitizable ordinals occurring much more often than non-subitizable ones. As was discussed in Chapter 6, ordinal frequency in extant languages decreases significantly at fourth and higher, even when the language has unrestricted numbers, a well-developed lexicon for number words, and vocabulary for non-subitizable ordinals like seventieth, hundredth, thousandth, billionth, and trillionth. The correspondence between usage frequency and the number sense is probably not a coincidence, since ordinals like first, second, and third are probably more salient as a function of their subitizability and relative social importance, like the top three contenders in an Olympic contest, who receive the gold, silver, and bronze medals. In contrast, the ordinals fourth and higher, though lexically expressible, may be conceptually less distinguishable among the many of magnitude appreciation. Comparing ordinal frequency in Sumerian to that of extant languages is complicated by several factors. Corpora for extant languages are realized by data mining extensive bodies of text; word frequencies are then calculated on hundreds of thousands to several million words. In contrast, the Sumerian corpus is compiled from the relatively few texts that have been found, translated, and published, yielding a much smaller list of words with fewer samples per item. Modern scripts are produced by literate societies using writing elaborated over millennia to become highly expressive of language; by comparison, the Sumerian corpus also represents a writing system undergoing significant elaboration to develop expressiveness for a significant amount of the time considered. There are also considerable differences in the coverage timespans: Modern corpora cover days to years, while the Sumerian corpus spans hundreds and thousands of years, depending on how much of it is included. The document composition differs as well, with modern corpora emphasizing narrative texts like newspapers and journal articles and the Sumerian corpus including significant numbers of administrative documents more akin to spreadsheets and grocery lists. Because of the differences between modern and ancient corpora, I examined Sumerian ordinal frequency by time period (Table 7.4). Breaking the sample into smaller periods decreased the sample size even further and did little to mitigate the time spans involved, since these remained at the level of centuries. However, separating the sample by time period did help me examine the effects of expressiveness in the writing system (Fig. 7.4) and related phenomena like document composition (Fig. 7.5). Despite the small sample size and other differences between ancient and modern corpora, Sumerian ordinal frequency was found to be roughly analogous to that of English, Mandarin, and Arabic. This was particularly true for the literary corpora, the document composition most directly comparable to that of modern corpora.

7. ANCIENT LANGUAGES AND MESOPOTAMIAN NUMBERS

Ordinal number 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 12th 15th 18th Administrative Royal/monumental School texts Letters Legal texts Literary texts Lexical texts Mathematical Unspecified

ED IIIb

OA

BCE

BCE

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Ordinal Frequency by Time Period Lagaš II Ur III Early OB Old Assyr.

OB

2500–2340 2340–2200 2200–2100 2100–2000 2000–1900 1950–1850 1900–1600

2

100.0%

BCE

6 8 3 3 3 2

BCE

4 5 1

BCE

2131 2236 2850 2904 1015 1861 1755 1764 549

BCE

18 20 6 12 7 14 10 11 1

BCE

205 23 20 5 9 5 4

Document Composition by Time Period 96.0% 100.0% 99.8% 99.0% 9.6% 4.0% * 1.0% * * 3.7% * 4.1%

*

50 86 68 63 67 53 70 23 17 2 2 1

82.7%

28.7% 2.0% 0.6% 17.5% 0.6% 46.2% 1.8% 2.2% 0.2%

Table 7.4. Sumerian ordinal frequency by time period and document composition. Ordinal numbers first appeared in the Early Dynastic IIIb period. By the Old Babylonian period, high ordinals like fifteenth suggest the influence of administrative cycles, comparable to modern paydays. For the Ur III period, asterisks mark negligible amounts. Key: ED, Early Dynastic; OA, Old Akkadian; OB, Old Babylonian. The data in the sample were drawn from the database of the Cuneiform Digital Library on 5 October 2015.

To examine the effect of change in the expressiveness of writing on ordinal frequency, I compared Old Akkadian period as the earliest with sufficient data to the Old Babylonian period (Fig. 7.4). The Old Akkadian corpus is composed almost exclusively of administrative texts, while the Old Babylonian corpus contains a wide variety of genres. In both, ordinal frequency was roughly similar to that of modern languages, with the lowest ordinals having the highest frequency of use and higher ordinals having a decreasing frequency of use. The distribution shows an unusual higher frequency for second than first, and significant frequency for the ordinals third to seventh. These features may be artifices of the data, things like the relative paucity of exemplars and the state of the writing system, or an influence from administrative texts.

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Fig. 7.4. Sumerian ordinal frequency by writing state. (Left) The Old Akkadian period represents writing with little fidelity to language and is composed almost exclusively of administrative texts. It was selected to represent the low-fidelity state because of the paucity of data from the Early Dynastic IIIb period. (Right) The Old Babylonian period represents writing with relatively high fidelity to language and includes a wide variety of textual genres. Both graphs show the same general trend for ordinal frequency as that found in extant languages: The highest frequency of use occurs for the lowest ordinals, and there is a decreasing or negligible frequency of use for higher ones. The data for the two time periods are those shown in Table 7.4. Image created by the author.

Finally, to see whether document type had an effect, I compared ordinal frequency in the literary texts to ordinal frequency in the administrative ones (Fig. 7.5). This lumped together these document types from all of the time periods, resulting in extremely long coverage periods. As there were negligible data to add from the Middle Babylonian, Neo-Babylonian, and Neo-Assyrian periods, the graph for the most part reflects the data presented in Table 7.4. Both distributions were characterized by a higher frequency of word use for second relative to first, as well as elevated usage, relative to modern corpora, between fourth and seventh or eighth. For the administrative texts, ordinal frequency most closely resembled that of the Ur III period. For the literary texts, ordinal frequency somewhat resembled that of the Arabic corpus. However, in both cases, caution again must be exercised in interpreting the results because of the various limitations of the Sumerian sample.

Fig. 7.5. Sumerian ordinal frequency by document type. (Left) Ordinal frequency for literary texts in the Old Babylonian (1900–1600 BCE; n = 232, 98.3%), Middle Babylonian (1400–1100 BCE; n = 3, 1.3%), and Neo-Babylonian periods (626–539 BCE; n = 1, 0.4%). (Right) Ordinal frequency for administrative texts as shown in Table 7.4, adjusted by adding data from the Neo-Assyrian period (911–612 BCE; n = 2, 0.01%). The data for the two document types were compiled from Table 7.4. Image created by the author.

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The main takeaway from the analysis were these: First, because even the earliest Sumerian ordinal numbers generally follow the tendency to use small ordinals, they are not a reliable measure of lexical extent and therefore cannot serve as the basis for classifying Sumerian numbers as restricted. Second, because they were formed like ordinals in extant writing systems by adding a sign for a sound value to a number sign, they show that the Sumerians formed their ordinals just like we do, not just in writing but also likely in speech. Finally, they also attest to the existence of the Sumerian numerical lexicon, since 15th implies fifteen.

WRAPPING UP THE EVIDENCE OF THE ANCIENT LANGUAGES Where have we gotten to with this examination of lexical numbers, grammatical number, and ordinal numbers in the Sumerian, Akkadian, and Elamite languages? The goals, after all, were to find evidence of finger-counting; establish that the populations who spoke these languages counted and invented their numbers like peoples do generally; show that these numerical traditions likely developed independently of one another; and characterize the timelines for their development. There is evidence of finger-counting in the overt compounding of the Sumerian lexical numbers six, seven, and nine, evidence of body-counting in the compounding of Sumerian lexical numbers ten through forty, and evidence of using the hands to count in the decimal organization of Akkadian and Elamite lexical numbers. This shows that ancient peoples inventing and using numbers had the same behaviors of counting with their bodies and using smaller numbers to build larger ones as extant peoples do. This evidence doesn’t give us a timeline, however, because the linguistic evidence cannot do this on its own and because there is no archaeological evidence to tie it to an absolute chronology. However, given the neurological evidence reviewed in Chapter 4 and the behavioral evidence reviewed in Chapter 5, finger-counting was likely to have been first, a relative chronology. If tokens were indeed used for counting as early as the 10th millennium BCE, then to have preceded it, finger-counting would have taken place in the Late Upper Paleolithic. Similarly, if artifacts resembling tallies did represent counting, as will be discussed in Chapter 8, the fact that some of these artifacts date to the Late Upper Paleolithic would push finger-counting back even further. The likelihood that the Sumerians, Akkadians, and Elamites had the same perceptual experience of quantity we do was supported by finding the structure of their grammatical number and the word frequency of their ordinal numbers conforms generally to the Weber–Fechner constant, as is the case for extant languages. We also saw that their lexical numbers were subject to the same kinds of processes— grammaticalization, lexicalization, and linguistic change—found in extant languages. The Sumerians also built their ordinal numbers in writing in the same way seen in extant languages, by adding a sign for a sound to a written number. This means they likely formed their spoken ordinals in the same way, by modifying a spoken number with an extra sound, like we do in English by adding -th or similar to the end of a number word. These findings, in conjunction with the evidence that they counted with their fingers, shows that the ancient peoples, more likely than not, counted and invented their numbers just like we do. This, in turn, argues against the idea that Mesopotamian numbers were exceptional, as they would have to be if they were as concrete as a Piagetian interpretation makes them.

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Differences in numerical organization—sexagesimal and ternal for the Sumerians, decimal for the Akkadians and Elamites—suggest these numerical traditions developed independently, as do the lack of any commonality between (non-borrowed) Sumerian and Akkadian lexical numbers and the variation in the aspects of grammatical number in Sumerian, Akkadian, and Elamite. Ordinal numbers suggest the Sumerian number system was unrestricted but leave unresolved the question of its developmental timeline. Grammatical number suggests lexical numbers, at least to the restricted level, had developed by the start of the Neolithic, placing their origin in the Late Upper Paleolithic, a timeline not incommensurable with the longevity discussed for Proto-IndoEuropean numerals in Chapter 6. The unanalyzability of the words for the numbers one through ten in Akkadian but not Sumerian suggest Akkadian, a Semitic language, was the oldest of the numerical traditions. Proto-Semitic has been estimated as originating around 3750 BCE in the Levant,62 a timeline that may be too shallow for the linguistic change demonstrated by the Akkadian lexical numbers. Their roots perhaps lie in Proto-Afroasiatic, which is ancestral to Proto-Semitic and was spoken in the Levant well before the Neolithic.63 Reconstructed numbers for Proto-Semitic and Proto-Afroasiatic fall at the lower end of the restricted range.64 While it is tempting to infer a restricted lexical range—or, at the other extreme, unanalyzable syllables—the paucity of data means caution is warranted in drawing any conclusions beyond the fact that a precursor numerical vocabulary existed on a timeline likely to have produced the pattern found in Akkadian lexical numbers. Conversely, the overt compounding, restricted range, and limited potential for productivity of the lexical numbers in the Sumerian ternal counting system suggest it was the youngest of all the indigenous numerical traditions in the Mesopotamian melting pot. Now that we have established that ancient peoples behaved with numbers like we do, talked numerically like we do, and had the same perceptual experience we do, a further inference is possible: what demographic factors suggest about a timeline for developing unrestricted numbers. Numbers are a response to—and thus, an indication of—social needs for controlling resources, especially in the face of increasing scope and scale. We can draw upon the well-known relation between socio-material complexity and numerical elaboration: The ‘elaboration of overall complexity in numeral systems—i.e., the development from highly restricted to generative systems that can name precise quantities into the tens, hundreds, or thousands—is motivated (at least in part) by social and cultural practices that encourage counting and/or keeping track of precise quantities’.65 Although a general relation between socio-material complexity and numerical elaboration has been established, scholars have been relatively unsuccessful in finding 62 Kitchen et al., ‘Bayesian Phylogenetic Analysis of Semitic Languages Identifies an Early Bronze Age Origin of Semitic in the Near East’. 63 Blažek, ‘Afroasiatic Migrations: Linguistic Evidence’; Ehret, Reconstructing Proto-Afroasiatic (Proto-Afrasian): Vowels, Tone, Consonants, and Vocabulary. 64 Ehret, Reconstructing Proto-Afroasiatic (Proto-Afrasian), pp. 109, 228, 273–274, and 424. 65 Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’, p. 42.

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specific correlations between numerical elaboration and things like subsistence patterns, food storage, trade, division of labor, and climate.66 My own initial foray into the matter found a correlation between socio-material complexity, numerical elaboration, and the use of material devices for representing and manipulating quantity.67 Simply, the development and elaboration of lexical numbers corresponded to increases in possessions and resources needing management, the emergence of value systems motivating the enumeration and accounting of goods, and the invention and use of material technologies for counting and record-keeping. Development and elaboration were also motivated by and intensified with contact between groups, increasing group size, or both. In hindsight, beyond noting that—of course—material devices for counting are just the thing for counting to higher numbers, the question of why socio-material complexity would mean the increased use of material devices for counting was left unanswered. This is something I hope to have answered by the final chapters of this book. Here it will suffice to say that in general, restricted number systems are associated with small-scale, traditional societies living in relative isolation.68 This condition obtains in the Ancient Near East before the Neolithic, which is also when grammatical number would have developed independently. Number systems counting into the hundreds or perhaps thousands are associated with larger societies living in villages in conditions of periodic contact and trade with other peoples. This condition appears with early Neolithic settlements around 8500 BCE and village communities by 7000 BCE.69 Number systems counting into the thousands, tens of thousands, and beyond are associated with large-scale, urbanized societies with well-established trade and significant internal complexity to manage.70 This is the condition of Mesopotamia by the mid-to-late 4th millennium BCE, with its intensive agriculture, monumental architecture, and complex bureaucracy; the same pattern occurred in Egypt, China, and Mesoamerica.71 66 Divale, ‘Climatic Instability, Food Storage, and the Development of Numerical Counting: A Cross-Cultural Study’; Hammarström, ‘Restricted Numeral Systems and the Hunter–Gatherer Connection’; Heine, Cognitive Foundations of Grammar, pp. 24–25; Stampe, ‘Cardinal Number Systems’, p. 596; Winter, ‘When Numeral Systems Are Expanded’, p. 43. 67 Overmann, ‘Material Scaffolds in Numbers and Time’; Numbers and Time: A CrossCultural Investigation of the Origin and Use of Numbers as a Cognitive Technology. 68 Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’. 69 Bar-Yosef, ‘On the Nature of Transitions: The Middle to Upper Palaeolithic and the Neolithic Revolution’. 70 Chrisomalis, ‘The Cognitive and Cultural Foundations of Numbers’; Divale, ‘Climatic Instability, Food Storage, and the Development of Numerical Counting’; Epps et al., ‘On Numeral Complexity in Hunter–Gatherer Languages’; Greenberg, ‘Generalizations about Numeral Systems’; Heine, Cognitive Foundations of Grammar; Winter, ‘When Numeral Systems Are Expanded’. 71 Chrisomalis, ‘Evaluating Ancient Numeracy: Social versus Developmental Perspectives on Ancient Mesopotamian Numeration’; Numerical Notation: A Comparative History; Damerow, Abstraction and Representation: Essays on the Cultural Evolution of Thinking; Houston, ‘Mesoamerican Writing’; Ronan and Needham, The Shorter Science and Civilisation in China.

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Given the urbanization and resourcing practices of the mid-to-late 4th millennium BCE, Mesopotamian peoples should have been capable of counting to the tens of thousands or higher, levels consistent with their material expression of numbers as numerical impressions and proto-cuneiform numerical signs. Counting systems capable of counting into the hundreds or perhaps thousands would be predicted between the 9th and 5th millennia BCE, in conjunction with increasing sedentism and food cultivation of the Neolithic.72 This corresponds well to the numbers associated with token finds from that period and is addressed in Chapter 9.73 Finally, the emergence of initial number concepts is likely to have occurred even earlier, in conjunction with hunter–gatherer modes of existence in the Late Upper Paleolithic.74 This is consistent with finds of possible tallies from that period in the Levant and the later linguistic evidence of numerical language. Taken altogether, the linguistic evidence provides a unique window into Mesopotamian numbers. But though informative, the linguistic evidence alone cannot tell us everything we want to know about ancient numeracy. For this we must turn to the archaeological record, the subject of the next several chapters.

72 Lloyd, The Archaeology of Mesopotamia; Nashli and Matthews, The Neolithisation of Iran: The Formation of New Societies; Reade, Mesopotamia; Schmandt-Besserat, ‘The Emergence of Recording’. 73 Cuneiform Digital Library. 74 Coinman, ‘Worked Bone in the Levantine Upper Paleolithic: Rare Examples from the Wadi Al-Hasa, West-Central Jordan’; Gladfelter, ‘The Ahmarian Tradition of the Levantine Upper Paleolithic: The Environment of the Archaeology’; Kuhn et al., ‘Ornaments of the Earliest Upper Paleolithic: New Insights from the Levant’.

CHAPTER 8. FINGERS AND TALLIES As we saw in the last chapter, there were at least three and possibly as many as five number systems coming into contact with each other in Mesopotamia during the Neolithic and later periods. These numbers differed in ways that suggest they originated independently. They also differed in their respective ages, with Akkadian likely representing the oldest and ternal counting perhaps the youngest of the five. These numbers were realized through the same behaviors and psychological processes found in extant number systems today, as attested by evidence of finger-counting and the characteristic limits of the number sense in linguistic forms of number. Such linguistic evidence requires a system of writing be invented and then elaborated to the point where it can express things like grammatical number and the spoken form of lexical numbers, and thus, it falls much later than what it reflects, linguistic forms of number that would have developed before the populations were much in contact with each other, prior to the Neolithic. Number words in the ancient Sumerian, Akkadian, and Elamite languages, in turn, were subject to the same frequencies of use and processes of change as number words in extant languages, and these processes and the resultant characteristics attest to both the respective ages of the various numerical traditions and the commonality of their psychological, behavioral, and material origins. Establishing that the peoples of the Ancient Near East perceived quantity, counted with their fingers, and spoke numerically the way we do today is important. First, it dispels the notion Mesopotamian peoples were unusual or atypical in their numbers, being particularly concrete in their thinking when they represented their numbers with tokens or necessarily using tokens as their first material technology for numbers or without a concomitant numerical vocabulary. It also warrants our assuming, even if we can’t see it archaeologically or textually, the various Mesopotamian numerical traditions would have included an earliest stage similar to one found in emerging systems today, in which people appreciate and represent subitizable quantity iconically and indexically, like the Mundurukú’s recognition that human arms exemplify two-ness and recreating two-ness with sound and gesture.1 In Mesopotamia, as is the case for other peoples today, this earliest stage would have started long before they transcended the subitizing range by counting the fingers of one hand and began patterning their numbers with the 1

Rooryck et al., ‘Mundurukú Number Words as a Window on Short-Term Memory’.

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fingers’ characteristics. Ternal counting is perhaps a glimpse of this earliest stage, speculation based on the resemblance of the compounding used in ternal counting and emerging numbers like those of the Bakaïrí (Chapter 5). Ternal compounding built a limited number of non-subitizable numbers in a way that didn’t overtly involve counting on the fingers, at least not with productive groups of five and ten like those of Sumerian lexical numbers. Since ternal counting was contemporaneous with the other, more elaborated Mesopotamian number systems, it was more likely to have been a minority numerical tradition (Chapter 7) than a preserved state of early Sumerian lexical numbers. I won’t mention this earliest step into numeracy further, beyond noting there would have been one, as my focus here is on the earliest material forms used for counting in Mesopotamia, fingers and tallies. Finger-counting is attested by overt fiveplus compounding in the Sumerian lexical numbers six, seven, and nine, and as the decimal organization of Akkadian and Elamite numbers. Tallies, as we’ll see, are attested archaeologically, as counting devices in Mesopotamia from the 1st millennium BCE and as artifacts possibly used for counting in the Levant during the Late Upper Paleolithic; tallies are also attested textually, as they are mentioned in a poem from the 2 nd millennium BCE. Both fingers and tallies are capable of accumulating, giving them the potential to help realize ordinal relations between numbers in a sequence (recall Fig. 5.1). Because fingers are naturally grouped by fives and tens, depending on how peoples count on their digits, and by twenties if they include the toes, they also have the potential for grouping relations between numbers organized as bundles (Fig. 5.2). Neither device necessarily represents the same concept of number found in the highly elaborated Western numerical tradition, where numbers are considered entities of an abstract and somewhat mysterious nature (Chapter 3). Instead, when represented by the fingers, numbers might be equivalences of quantity between sets of objects, like five of something has the same quantity as the fingers on one hand. When represented on a tally, numbers might be collections corresponding, one to one, to whatever it is they count, as one mark counts one of something in a set of objects and all the marks count the whole set. Since I keep mentioning concepts, this might be the point to ask, what’s a concept? Considered as something that happens internal to the brain, the answer is—no one really knows, since we don’t understand how or why neural activity gives rise to phenomena like consciousness, experiential quality, and meaning. 2 The idea that concepts are something the brain does is also mental representationalism, something Material Engagement Theory (MET) specifically rejects, though I also think it’s wise to leave the door open just a crack to admit cognitive states like reverie and dreams that do not explicitly involve interactions with material forms. The answer to ‘What’s a concept?’ becomes marginally more satisfactory when we consider it something that emerges from a system composed of brains, bodies, and material forms, since this framework relieves the brain from being solely responsible for generating concepts and lets us 2 Chalmers, ‘Facing up to the Problem of Consciousness’; Jackson, ‘Epiphenomenal Qualia’; Searle, ‘Minds, Brains and Programs’.

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trace conceptual properties to material ones. Admittedly, the non-representational perspective still doesn’t answer the how and why questions of what a concept is. However, as I noted in the introduction, solving such difficult and long-standing issues isn’t my goal; rather, I want to understand how peoples use material forms to realize concepts of numbers, and how those concepts change when we incorporate new material forms. These things are tractable to being examined and analyzed, even if we aren’t really sure what a concept is or what the brain does when we have one. For regardless of whether concepts are brain-internal or systemically realized phenomena, they have external aspects we can observe and investigate. For ancient peoples, this means examining the material forms they used for numbers and defining the properties of these devices in order to understand their influence on the content, structure, and organization of the concepts we call numbers. Some of these properties are explicitly encoded in the material forms themselves. Others take the form of knowledge, norms, and habits. These are mental and behavioral matters acquired by interacting socially or experiencing other material forms, as described in Hutchin’s model of conceptual blending,3 and they influence how we use material forms and understand the concepts they encode and transmit over space and time.4 From the material forms themselves, some behaviors are visible, though knowledge and norms tend not to be. If ‘all the ingenuity in the world will not replace the evidence that is lost and gone for ever’ and we ‘should recognize [our] guesses for what they are’,5 ‘inferring intelligent behavior from its material relics’6 is nonetheless something only archaeology can do. We can infer some knowledge and norms from the material properties of the devices used for counting, given that such devices are in use today and we have established that both ancient and extant peoples share behaviors and psychological processing in numbers. Keep the model and the analytical strategy in mind as we consider the sequence of material forms used for counting in the Ancient Near East, starting with fingers and tallies.

THE WHY, HOW, AND WHAT OF FINGER-COUNTING Let’s start with why we use material forms of any type for numbers. It’s not just that material forms have a quality we perceive with our number sense and thus instantiate that quality in ways we can exploit when we use them to represent and manipulate it. Importantly, material devices also help us remember things in ways that exceed the capacity of our human memory, a function called externalized storage for mental content in discussions of prehistoric artifacts.7 We tend to think of externalized storage in terms of significant data and long spans of time—in modern parlance, as tallies’- and books’Hutchins, ‘Material Anchors for Conceptual Blends’. Hutchins, Cognition in the Wild. 5 Leach, ‘Concluding Address’, p. 768. 6 Renfrew, Towards an Archaeology of Mind: An Inaugural Lecture Delivered before the University of Cambridge on 30th November 1982, p. 4. 7 d’Errico, ‘Memories out of Mind: The Archaeology of the Oldest Memory Systems’; Donald, Origins of the Modern Mind: Three Stages in the Evolution of Culture and Cognition. 3 4

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worth of numbers over hours to days to years, matters where their role in preserving and transmitting cultural knowledge is quite apparent. Their role in storing small amounts of information momentarily tends to be less remarked. In the moment, material forms anchor and stabilize concepts like numbers.8 This makes it easier for us to use numbers in processes like counting, first, because there is a material form representing numbers that we can see, touch, and move, and second, because they persist in what they represent. These are qualities words can’t as easily provide. Until writing is invented, words can neither be seen nor touched, and they cannot be moved once expressed. In functions like holding and manipulating, words are kept in working memory, where they are subject to the limit on the number of items we can remember and manipulate mentally, which has been thought for several decades now to be about seven items, plus or minus two.9 To be retained in working memory, words generally need to be subvocally rehearsed—repeated mentally, something you might do to remember a telephone number between looking it up and dialing it. Even then, words in working memory are subject to interruption, alteration, decay, and forgetting, effects that challenge our short- and long-term memory functions as well. In recognizing their role in extending human memory, historical discussions about externalized storage have tended to foster an impression that material forms used for counting are passive recipients of the mental content they display. When they are viewed through MET, however, they are seen instead as actively contributing to human numeracy, co-constitutive of the human cognitive system for numbers along with the psychological, physiological, and behavioral capacities they interact with. When we manipulate the quantities they instantiate, material forms keep track of the accumulations and other combinations they form in being rearranged, enabling the complexity of our calculations to increase. They also instantiate the patterns and relations that occasion our recognition and spark our insight, giving our brains something to do it’s really good at, which is recognizing and completing patterns. It’s important to note that using material forms for counting and calculating doesn’t always mean using the fingers or a device like a tally, since the objects themselves can track and display an accumulated quantity as we rearrange them, as we saw was the case for Polynesian counting. Objects rearranged in counting distinguish their quantity to us visually, identify which of their group have been counted and which have not, and, tally-like, track their own quantity as it accumulates. These functions, in turn, decrease the demands on cognitive processes like attention and working memory. We use the fingers in particular as a material form for numbers. This is because of the interaction between the intraparietal sulcus, the part of the brain that appreciates quantity, and the angular gyrus, the part that ‘knows’ the fingers and is involved in higher-level, cross-domain thinking and metaphorizing (Chapter 4). This neural under8 Hutchins, ‘Material Anchors for Conceptual Blends’; Malafouris, How Things Shape the Mind: A Theory of Material Engagement. 9 Baddeley, ‘Is Working Memory Still Working?’; Baddeley and Hitch, ‘Working Memory’; Miller, ‘The Magical Number Seven plus Minus Two: Some Limits on Our Capacity for Processing Information’.

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pinning makes using the fingers for numbers cognitively prepotent and positions the fingers as a bridge between internal psychological processes that appreciate quantity and the external physical world where things like hands have quantity. Here the hands are both actor and instrument,10 enabling us to rearrange the things we count and acting themselves as a material structure for representing and accumulating quantity. This neurological underpinning also helps us express our embodied experience of quantity with gesture, both physically and linguistically. The fingers make a handy tool for transcending the subitizing range, not just readily available but easily intuited, since the meaning of concepts like ‘as many as the fingers on one hand’ leverages, and is anchored and stabilized by, the shared embodied experience and appreciated quantity of the fingers. Peoples also tend to use the fingers for counting across the full spectrum of numerical elaboration, as it occurs not just with emerging, initial numbers but also ones long established and highly elaborated. This persistence is consistent with, and likely explained by, finger-counting’s neurological substrate; individual/communal availability; their appreciability and commonality; and their ability to decrease demands on, and thereby extend the capacities of, our cognitive resources. How we use material forms in numbers, in the broadest sense, can be categorized as two main functions: representing, which includes things like displaying, visualizing, storing, recalling, and transmitting, and manipulating, which includes organizing, transposing, and transforming. What societies need to do to survive their environments motivates them, or not, to incorporate and use material forms to perform these functions. It’s worth noting that few material forms fulfill both functions equally well. A tally is good at representing quantity but bad at manipulating it, since marks are difficult to move or remove once they have been made, while an abacus and a calculator are both good at manipulating quantity but bad at representing it, since display precludes the device from being used for manipulation for the duration. This is true of written notations as well. If we don’t think of notations as being limited in this regard, it’s only because we’ve developed strategies like crossing out and rewriting notations. We use these strategies with algorithms that decompose complex problems into series of small mental judgments, whose output values form the basis for replacing input values by crossing them out and rewriting them. The material form permits an interactivity that obscures the fact that written notations are in actuality fixed. Because it’s difficult to find a single device that fulfills both functions to the desired level of performance, societies tend to use one device for representing and another for manipulating, as notations were used for representing numbers, abaci and tokens for manipulating them, in Mesopotamia,11 as well as Europe.12 In terms of these two functions, we use the fingers to represent quantity for relatively short periods of time and manipulate it in the limited sense of displaying a range of numbers that typically fall between one and five and performing simple operations Gallagher, ‘The Enactive Hand’. Høyrup, ‘A Note on Old Babylonian Computational Techniques’, p. 2. 12 Reynolds, ‘The Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators’; Stone, ‘Abacists versus Algorists’. 10 11

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like accumulating. Given the preponderance of number systems containing number words that refer to the fingers and/or hands or using productive cycles based on fives and tens, most societies use the fingers for these functions. In realizing initial numbers, the fingers make what we perceive through our number sense tangible, persistent, and manipulable in ways that enable us to form concepts, use and share them socially, and name them in language. We also use the fingers to display quantity to other people, something known as finger-montring, from the French verb montrer, meaning to show or display. Finger-montring patterns often differ from those used to accumulate quantity. When counting to accumulate, I lower my thumb to the desktop for one, my index finger for two, my middle finger for three, my ring finger for four, and my little finger for five. But when I display these same quantities to another person, I raise my index finger for one; my index and middle finger for two; my thumb, index, and middle finger for three; all four fingers for four; and all four fingers plus my thumb for five. The difference between these functions can be explained as follows: The first helps me keep track of something I’m counting in a way that minimizes the requirements for attention and working memory. The second displays patterns that are visually distinctive to someone else. Because some of the patterns in finger-montring are more difficult to produce biomechanically than others,13 and because the fingers are used non-sequentially, finger-montring is somewhat costlier in terms of attention and working memory. When we consider the effect(s) of finger-counting on numbers, we need to consider what the hand offers as a material form used for representing and manipulating: its properties or affordances. In Chapter 2, I said an affordance was a property of the interaction between an agent’s abilities and its environment.14 Simply, affordances are properties of a material form that enable us to do something with it, provided we have the capacity—behavioral, physiological, and/or psychological—to exploit them. In other words, what a material form is influences what we can do with it; when a material device enables us to do something, its properties influence the results we achieve and the concepts we form. This definition can be further refined: As a device for representing and manipulating quantity, the hand has capabilities and limitations that affect why and how we use it for these purposes, and these qualities also influence the resultant concepts. What are the hand’s affordances with respect to numbers? First, as a material form, the fingers have quantity. This quantity exceeds the subitizing range of our perceptual experience. Moreover, the hands are positioned within our visual field, and they are constantly brought to our notice as we use them. Their quantity is visually appreciable, a quality that reinforces our embodied understanding of their quantity, even though it exceeds the subitizing range. The fingers, at the neural level and specifically in regard to quantity, interact with the number sense, making us predisposed to use the hand to express quantities we can both see and feel. Also at the neural level, the fingers’ movement and sensation is topographically organized in a linear fashion, so when we start using the hand to accumulate quantity, our numbers are influenced toward 13 14

Lin, Wu, and Huang, ‘Modeling the Constraints of Human Hand Motion’. Gibson, ‘The Theory of Affordances’, The Ecological Approach to Visual Perception.

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linearity. This visual linearity differs from the linearity of language: While sound is expressed in a linear stream, making it sequential, auditory sequences do not form straight lines, which is a material quality appreciated visually. The influence toward material linearity is modulated by the visually appreciated features of the hand, such that individuals and communities make different choices as to where to start and stop and how to proceed between the two. Also visually appreciated is the quality of discreteness, where numbers whose range is somewhat fuzzy become defined against one another when represented by the fingers. And of course, because our limbs are pentadactyl, using them for counting strongly patterns our number systems by fives, tens, and twenties, grouping as well as accumulating. We can also look at five principles of number systems considered to be invented, rather than innate15: The one-to-one principle assigns distinct representations or words to each item counted. The stable-order principle means that order is fixed. The cardinal principle is the idea that the number for the last object in a set represents the set’s quantity or cardinality. The abstraction principle states that numbers can be applied to any type of object to be counted. The order-irrelevance principle means that numerical value is not based on the order in which things are counted. These principles can be correlated with the affordances of the hand, as follows: The fingers are an external material structure connected to an internal sense of quantity that appreciates quantities shared by fingers and objects. Counting objects one by one on the fingers individuates fingers, objects, specific quantities, and their verbal names. Stable order or ordinality emerges next16 through the physical structure of the hand, which provides logical initiating and concluding points and a linear succession between them. Cardinality then emerges through the use of cumulative fingers to represent the total number of objects in a set. However, the ordinality of finger-counting may inhibit the emergence of the abstraction and order-irrelevance principles,17 perhaps necessitating the incorporation of material artifacts other than the fingers into the cognitive system for numbers, something motivated, or not, by social needs and purposes for counting. As a social behavior, finger-counting has several universal aspects to it—things all societies and people do when they finger-count. We all count on our fingers, for example, and this is independent on how elaborated our numbers happen to be.18 Within any particular society, people tend to count on their fingers the same way; the behavior is culturally transmitted and reproduced. It is also culturally inflected: Different societies count on their fingers differently. However, all finger-pattern variants involve establishing habitual start and stop points, regularizing the sequence between the two, and repeating the patterns to the point they become automated. Why this is so is apparent when the alternatives are considered: Random finger-patterns are inconsistent Gelman and Gallistel, The Child’s Understanding of Number. Andres, di Luca, and Pesenti, ‘Finger Counting: The Missing Tool?’. 17 Beller and Bender, ‘Explicating Numerical Information: When and How Fingers Support (or Hinder) Number Comprehension and Handling’. 18 Overmann, ‘Material Scaffolds in Numbers and Time’; ‘Finger-Counting in the Upper Palaeolithic’. 15 16

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and unreliable in representing and recovering numerical information. Non-sequential finger-patterns are more awkward to produce, biomechanically speaking, and less distinguishable visually. Both increase requirements for resources like attention and working memory. By comparison, always starting with the same point, proceeding in the same fashion, and ending at the same point facilitate biomechanical production, improve visual distinguishability, and minimize demands on cognitive resources, thereby improving the consistency, reliability, and recoverability of the information they represent. Habituation, where behaviors like finger-counting become customary, and automaticity, where behaviors are performed without much conscious thought, reduce the burden on cognitive resources even further. Our tendency to make use of the natural groupings of the hands and feet in counting is also universal. We involve the second hand once we have used all the fingers on the first; we may engage the feet when we have exhausted the fingers of both hands, though we might also omit the feet from counting altogether; and we recruit the digits of another person when we have used up all those of our own. This is not to say we invariably involve the next group at five or ten, just as we don’t always start and end in the same place or use the same features of the fingers to count on. But it does give us a way to realize the second exponential dimension, not just accumulating but grouping, and it inflects our number systems by digital amounts, usually but not invariably fives, tens, and twenties. This is not mere happenstance: Using these natural features not only decreases the demands on other cognitive resources, it also leverages bodily features and embodied understandings shared between individuals and across societies. This physically and communally anchors the meaning of terms like mijeŋ yagi, one-foot three in Sora, a language of southeastern India; tupesanpe ikasma wanpe, two-from [ten]and-both [hands] in Ainu, a language of Japan and eastern Russia; and duodēvīgintī, twofrom-two tens in Latin, a language of Mediterranean Europe, all of which mean eighteen.19 Another universal is that we count on our bodies in the same way we count in language. Simply, we tend not to count to ten on our fingers but use productive cycles of twelve in our number words, or vice versa; rather, we count to ten or twelve in both forms. This correspondence is partly a reflection of their historical relations: Number words correspond to the fingers because, quite often, they were once based on them, though this might not be apparent in cases where linguistic change has reduced bodybased phrases and words to unanalyzable syllables. It’s also partly that the two forms reinforce each other when used. If they differed, as might be the case in counting on the fingers with numbers that once included the toes or counting with number words borrowed from another language where they were counted on the body differently, this would work to bring the two forms into congruence. If they already were congruent, their mutually consistent structure would be reinforced each time they were used together. Again, it’s not difficult to understand why this is so: Mutually inconsistent structure would have many of the same drawbacks posed by random and nonsequential finger use.

19

Stampe, ‘Cardinal Number Systems’.

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We can apply these insights to Mesopotamian finger-counting. Formations like five and one that mean six in Sumerian lexical numbers, along with higher formations based on ten, suggest the Sumerians counted to five on one hand before switching and counting to five on the other, producing cycles of ten. This method of using the fingers to count is fairly common, cross-culturally speaking,20 likely because it leverages both the physical structure of the hand and the brain’s topographical layout for finger movement and sensation. The cycle of ten this method produces is consistent with the use of ten and twenty to form numbers like thirty and forty in Sumerian. It’s also consistent with the numerical organization of the Neolithic tokens, Chalcolithic impressions, and Bronze Age notations, where productive cycles of ten alternated with productive cycles of six to produce sexagesimal or base 60 numbers. That is, the decimal sub-base that underpins Mesopotamian sexagesimal numbers21 most likely originated as fingercounting. Cycles of five, ten, and twenty imply the Sumerians leveraged the natural groupings of the hands, feet, and people prior to the point when groupings of ten and six became standard; the analyzable compounding of their number words suggests they counted to about sixty before this occurred. Finally, the idea that number words and finger-counting are mutually consistent should dispel the curious notion that sexagesimal numbers were based on counting to five on the fingers and multiplying this cycle by twelve, or vice versa.22 While it’s true multiplying five by twelve produces sixty, cycles of five and twelve were not the basis of the Mesopotamian number words, tokens, numerical impressions, or notations; these were based instead on combinations of five, ten, and twenty or productive cycles of ten and six. Counting on the fingers most likely explains the decimal organization of Akkadian and Elamite numbers. These populations may have counted to five on each hand like the Sumerians did, given their cycles of ten and the prevalence of this method and its relation to human neurological and physiological structures. However, there is no overt compounding in their numbers six through nine to confirm this. For Akkadian, this is because the number words up to ten had become unanalyzable before they were preserved in writing, something that also suggests they were relatively old. For Elamite, it’s because pronunciations cannot be recovered from semasiographic notations, the only written form of these numbers currently known. The lack of overt cycles of twenty suggests both these peoples grouped their digits differently than the Sumerians did. Presumably, the Sumerians involved their feet before recruiting the digits of a second person, producing characteristic groupings of five and twenty, while the Akkadians and Elamites recruited the digits of a second person once they had exhausted the fingers of both hands, producing characteristic groupings of ten. This is similar to the difference between numbers in the Sora and Ainu languages mentioned earlier. It doesn’t imply the Akkadians and Elamites counted on their fingers the same way as each other, mere-

Overmann, ‘Finger-Counting in the Upper Palaeolithic’. Neugebauer, The Exact Sciences in the Antiquity, p. 19. 22 Huylebrouck, ‘Tellen Op de Handen in Afrika en de Oorsprong van het Duodecimale Systeem’. 20 21

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ly that they used a grouping strategy more similar to each other’s than to the one used in Sumerian numbers. It’s reasonable to assume that finger-counting would have had the same effects on ancient numbers as those found in extant number systems. Beyond five-based patterning, finger-counting would have influenced ancient numbers toward sequentiality, linearity, and discreteness, numerical properties emerging from properties of the hand as a material structure representing and manipulating numbers. These properties would have persisted both as the material form itself and in the forms of knowledge, norms, and habits. In turn, knowledge, expectations, and habits acquired from finger-counting would have influenced other material forms used for numbers as the need for additional forms, which specific forms were selected, and how the forms were used. For example, the fact the fingers aren’t good at storing numerical information for long periods of time would have motivated the use of material forms that could perform this function. Similarly, linearity acquired from using the fingers would have influenced the linearity found in numerical notations, as well as in the tokens used for counting, though these had a form whose linearity we can only assume before they were pressed into clay to form persistent linear impressions in the mid-to-late 4th millennium BCE. Finger-counting to the point where numbers are influenced and patterned by the properties of the hand implies a society that practices the behavior both communally and with enough frequency that habituation and automaticity occur. This implies both that a society needs numbers enough to invest in performing and sustaining the behavior and that it performs and sustains the behavior enough to constitute a mechanism for learning and transmitting number concepts between individuals and generations. This does not necessarily also imply that the society was large or complex. Certainly, finger-counting is well documented in small-scale, traditional societies with restricted number systems. Thus, it would also have been consistent with the demographic conditions of the Late Upper Paleolithic, which is when I estimate Mesopotamian numbers were most likely to have originated. It is to the archaeological evidence of this same time period I now turn.

EVERYTHING YOU WANTED TO KNOW ABOUT TALLIES In extant societies, tallies are often used side-by-side with the fingers for counting, and like finger-counting, their prevalence spans restricted to highly elaborated numbers. We use them ourselves, in fact, typically as marks of four vertical lines crossed by a fifth: ༛. Diachronically, tallies represent a shift from comparing things to the body in order to count them to comparing things to each other. This makes counting less of a private activity and more communal: Counting expressed in a material form avoids our crosscultural squeamishness about touching bodies publicly or imitating prestige behaviors. Because they are material, tallies also have an increased potential for distributing cognitive effort between individuals and across generations. Certainly, cultural knowledge and behaviors like finger-counting that use the body instrumentally are distributional mechanisms, but they have finite and relatively small capacities for encoding and elaborating information. Strictly speaking, tallies are also limited in their capacities for these things: As a material form, they are a relatively simple device, as measured by the resources needed to learn to use one or invent one from scratch. But because they are a material form that is not the body, tallies also represent the beginning of a powerful

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process—incorporating material forms for encoding and elaborating cultural knowledge—that for numbers begins with things like tallies and continues today with things like calculators and computers. If tallies are easy for novices to understand and reproduce, devices like computers are not, even for experts: At some point, the knowledge they encode and the affordances they offer exceed the capacity of what single individuals or generations can manage on their own. Tallies represent the first step in harnessing the power and agency of material culture to numerical purposes. What we find archaeologically is only a subset of the devices that can be considered tallies, the category of one-dimensional accumulators of quantity—things external to both the body and the things being counted—that includes things like notched pieces of wood and bone, knotted strings, torn leaves, stripes of body paint, marks on the ground, and piles of stones or grain. If they survive the process of deposition, we have to discover them, and this is more difficult when social groups are demographically small and isolated, like they were in the Late Upper Paleolithic and early Neolithic, since any material culture they leave behind them will also be small and isolated on the landscape. The likelihood of preservation and discovery decreases with the span of time. The little that does survive tends to be made of only the most durable stuff: stone, bone, and fired clay. We also might find but not recognize things like collections of stones or grain as having once been used for counting. After subtracting the things that don’t preserve and which don’t overtly look like counting devices, we are left with only the things made of highly durable materials in the specific form of artifacts whose surfaces are scored with linear marks. Most of the things described ethnographically either wouldn’t preserve or wouldn’t be recognized. Once a prehistoric artifact with incised marks survives the deposition, discovery, and recognition processes, we must determine whether it was, in fact, used for counting. It’s important to note that marks made for counting are typically regular and linear, likely a function of structuring from the previous use of fingers as a counting device, the matter of visually appreciated material linearity mentioned in the earlier discussion of finger-counting. Regularized linearity also ensures the represented information is more accessible and recoverable: Relative to marks that are haphazardly disposed, marks that are regular and linear represent quantity information more clearly, a quality that facilitates its subsequent appreciability and intelligibility. But as it happens, regularized linearity is also characteristic of marks made for other purposes and functions, like dividing a musical instrument into segments related to musical intervals. Simply, regularized linearity is not a reliable criterion for identifying a tally, and archaeologists consider a number of characteristics when assessing whether an artifact may have been used as one: x

Series of marks are assessed for consistency of length, spacing, and orientation.23 Consistency distinguishes marks made for purposes like decoration and enumeration from marks produced by processes like butchery, where marks

23 Marshack, ‘Evolution of the Human Capacity: The Symbolic Evidence’; The Roots of Civilization: The Cognitive Beginnings of Man’s First Art, Symbol and Notation.

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THE MATERIAL ORIGIN OF NUMBERS are haphazard as an inadvertent byproduct, and purposes like grip improvement, where marks might be intentional but irregular. Consistency also characterizes functions with little relation to counting per se, like making music, working fibers and leather, and divination.24 A specific type of consistency might suggest counting: A case where every fifth mark was longer or spaced more widely or oriented perpendicularly to the rest would suggest counting by fives. However, none of the prehistoric artifacts I know of are unambiguous in this regard. x

How marks are distributed across the artifact’s surface is described,25 as this might differentiate counting from purposes like decoration and work functionality. Marks filling an artifactual surface evenly suggest they were planned, which in turn suggests they were made for purposes like decoration and work functionality because the process of making marks was halted once the surface was filled. In comparison, marks bunching up imply a lack of planning, which in turn suggests counting because the process of making marks continued despite running out of room.

x

The number of marks is compared to known phenomenon. A group of 28 or 29 marks, for example, would imply the days in a lunar cycle, especially if it were further subdivided to suggest the different faces or light conditions of the new moon, first quarter, full moon and last quarter.26 If what was once counted was something cultural and thus arbitrary and conventional, like the price of a bride, the number of marks probably wouldn’t illuminate the purpose or function of the device for us today.

x

Wear patterns attest to an artifact’s use and curation. Extensive wear shows the device was kept and used for a long time, while little wear suggests the opposite. Both are consistent with tallies, since these are used both as ad hoc devices to assist particular counting processes and long-term devices with calendrical and record-keeping purposes. While extensive wear would tend to exclude ad hoc use and might suggest a repeated purpose like keeping track of time, it would also be consistent with purposes like making music and tool functionality.

x

Marks can be examined microscopically to determine whether they were made by the same or different tools, a now-classic analytical method pioneered by Archaeologist Francesco d’Errico and his colleagues.27 A single tool used by a

Reese, ‘On the Incised Cattle Scapulae from the East Mediterranean and Near East’. Marshack, ‘The Taï Plaque and Calendrical Notation in the Upper Palaeolithic’. 26 Emmerling, Geer, and Klíma, ‘Ein Mondkalenderstab aus Dolní Věstonice’; Marshack, The Roots of Civilization. 27 d’Errico, ‘Microscopic and Statistical Criteria for the Identification of Prehistoric Systems of Notation’, ‘Palaeolithic Origins of Artificial Memory Systems: An Evolutionary 24 25

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single individual in a single session produces marks with the highest degree of similarity, relative to marks made by different individuals, tools, or times, or combinations of these factors. A high degree of similarity, then, suggests marks made with the same tool at the same time by the same individual, and this in turn suggests they were more likely made for a purpose like decoration and work functionality, rather than counting. A high degree of dissimilarity, in contrast, suggests different tools, times, and individuals; this implies cutmarks made over time, a temporal distribution more consistent with enumeration. Figuring out what a prehistoric artifact was once used for is easier if lots of them are found, since they can be compared in characteristics that suggest their purposes and functions, like the material they were made of, their physical dimensions and weight, wear patterns, and the contexts in which they were found. However, prehistoric devices that look like tallies are relatively infrequent, with those older than about 10,000 years tending to be one-offs. Two of the oldest tallies are from Africa, and one is from Europe: x

The Lebombo bone is estimated to be more than 40,000 years old; it has 29 incised notches of roughly the same length, spacing, and orientation, and its wear, a glossy polish, shows it was curated for a long time.28 The artifact was said to resemble wooden sticks used as calendars by ‘some Bushman clans in South West Africa’,29 a claim widely repeated. However, photos and descriptions of the calendar sticks purportedly used by the San people remain unavailable for comparison, and in any case, culturally linking extant peoples to those inhabiting the same region some tens of thousands of years earlier is problematic.

x

The Ishango bone, also African, is younger at about 25,000 years.30 Its three rows of marks are differentiated by spacing and orientation into mathematically suggestive groups. The first row contains 11, 13, 17, and 19 marks, the prime numbers between 10 and 20; the second row has 9, 19, 21, and 11 marks, suggesting 1 plus and minus 10 and 20; and the third row contains 7, 5, 5, 10, 8, 4, 6, and 3 marks, suggesting halving and doubling, though not consistently.31 The seeming impossibility of such sequences being produced by chance, especially when taken altogether, has fueled speculation about mathe-

Perspective’; d’Errico et al., ‘Archaeological Evidence for the Emergence of Language, Symbolism, and Music—An Alternative Multidisciplinary Perspective’. 28 d’Errico et al., ‘Early Evidence of San Material Culture Represented by Organic Artifacts from Border Cave, South Africa’. 29 Beaumont, ‘Border Cave: A Progress Report’, p. 44. 30 Brooks and Smith, ‘Ishango Revisited: New Age Determinations and Cultural Interpretations’. 31 De Heinzelin, ‘Ishango’.

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THE MATERIAL ORIGIN OF NUMBERS matics having deep African roots.32 However, the device has a piece of quartz affixed to one end, a feature more consistent with engraving than counting. In addition, a second bone found in the same archaeological layer remains unpublished, presumably because its 90 marks are not mathematically interesting.33 x

An incised bone from Dolní Věstonice is about the same age as the two from Ishango.34 The artifact is typically described as having ‘55 notches set in groups of five’, sometimes but not invariably further specified as ‘25 notches separated from the other 30 notches by having one notch double in size from the others’.35 The logic of the claim is that the shorter sequence represents 5 × 5, the longer 6 × 5.36 In fact, photos show the device containing 55 marks in a single row, with the 25th and 26th being noticeably longer and the 47th being slightly longer than the rest.37 Not only is this less suggestive, mathematically speaking, the marks are specifically not grouped by fives the way our own tally marks are: ༛ Also unexplained is why the purported 5 × 5 sequence ends with a longer mark and the 5 × 6 sequence begins with one.

These prehistoric devices are isolated in both time and space, with the two closest, the ones from Lebombo and Ishango, still separated by some 17,000 years and 3800 kilometers. Other forms of material culture, things like stone tools, have also been found at these sites but tend not to provide clues to the functions and uses of the possible counting devices. Compounding the isolation problem is the fact we don’t really know when people began counting. While there are grounds for arguing counting could have started as early as 50,000 years ago at the beginning of the Late Stone Age in Africa and Late Upper Paleolithic in Europe, or perhaps even 100,000 years ago in the African Middle Stone Age, the arguments are inferential, referencing artifacts that may—or might not—have been used for counting. It’s a further inferential leap, not to mention circular reasoning, to claim that because counting could be that old, specific artifacts must have been used for counting. Hypothetically, let’s say all the devices from Lebombo, Ishango, and Dolní Věstonice were tallies, devices intended to represent counting. How would the information be used, subsequent to making the device? The quantities represented fall above the range of both subitizing and restricted numbers. Their number isn’t appreciable just by looking, so they would either represent approximated magnitudes—‘as 32 Evans, The Development of Mathematics throughout the Centuries: A Brief History in a Cultural Context, p. 4; Huylebrouck, ‘The Bone That Began the Space Odyssey’; Pletser and Huylebrouck, ‘The Ishango Artefact: The Missing Base 12 Link’. 33 Royal Belgian Institute of Natural Sciences, ‘The Second Ishango Bone’. 34 Oliva, Dolní Věstonice I (1922–1942): Hans Freising – Karel Absolon – Assien Bohmers. 35 Evans, The Development of Mathematics throughout the Centuries, p. 4. 36 Absolon, ‘Dokumente und Beweise der Fähigkeiten des Fossilen Menschen zu Zählen im Mährischen Paläolithikum’, p. 147. 37 Evans, ‘Three Views of a Wolf Bone Excavated at Vestonice’, p. 553.

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many as the notches on the stick’—or they would need to be counted again. Yet despite extending beyond the restricted range, the numbers of notches cannot guarantee the availability of a numerical lexicon, making counting a matter of comparing the notches again, one by one, to whatever it was they counted. This is easier when the notches are relatively large and widely spaced, like they are on the Lebombo bone. Days of a lunar cycle might be counted by moving a piece of twine to the next notch each day, similar to the way the Wathaurung messenger made and erased stripes of mud to count his days of travel.38 Such use might explain the Lebombo artifact’s glossy polish, though use as a tool or musical instrument has been proposed for other cases with such wear. Similar use might also explain the single row of 55 notches on the Dolní Věstonice artifact, though its notches are smaller and closer, making their potential reuse in this fashion more difficult. Its surface wear has not been described, at least not that I’ve been able to discover, and might not even be assessable, as the artifact was reportedly damaged in the 1945 fire at Mikulov Castle, where it was stored.39 As for the interesting marks on the Ishango bone, they are too small for incremental oneto-one comparisons. For their numerical information to be useful, the marks would need to be counted in language—not only a tedious prospect, given their size, but one presupposing both the availability of a numerical lexicon and an understanding of the mathematical relations and qualities purportedly depicted. Else, they would remain a personal code, perhaps imbuing the device with the magic of its maker’s knowledge. When we apply MET to the question of whether prehistoric devices were tallies or not, our focus shifts from discerning an artifact’s purpose to recognizing the cognitive opportunities it would have provided. That is, marks presuppose an interaction with a material form that changes it in some way. Making marks entails the performance of motor actions and some development of skill, as well as the kind of visual stimuli and potential regularities important in numerical patterning. If originally inadvertent, each mark is still an opportunity to make another one on purpose. If originally decorative or made to support a work function, each mark is nonetheless related to the others that precede and follow it, opportunizing concepts of how many, more than, less than, the same as, not the same as, and potentially, one more than. Thus, even if originally inadvertent or decorative in their intent or made as part of a tool, marks represent engagements with material forms that gave people opportunities for recognizing patterns and forming concepts. Because they are a material form separate from the body, tallies have affordances distinct from those of the body. Because they are separate from the body, tallies are less private to the individual, more public and shareable for communal use, and can be in places the body isn’t. They persist longer than representations of quantity made with the fingers, a quality modulated by the durability of what they are made of and the persistence of marks relative to processes that wear them away: Marks incised into bone or wood last longer than marks scratched on the ground or painted on the arm. Marks are 38 Matthews, referenced in Blake et al., ‘Wathawurrung and the Colac Language of Southern Victoria’. 39 Oliva, Dolní Věstonice I (1922–1942), p. 28.

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accumulative but not subtractive, as it is relatively easy to add more of them but difficult to move or remove them. This makes them more likely to scaffold the idea of addition than subtraction, and it’s consistent with the observation that cross-culturally, addition tends to emerge before subtraction does. Interestingly, the fingers don’t seem to support the idea of subtraction either, despite their early and persistent use and cross-cultural ubiquity; perhaps because they are part of the body, they cannot be removed completely or frequently enough to scaffold the idea of subtraction. As accumulators, tallies also have the potential to extend counting sequences to quantities beyond the capabilities of the hands, toes, and other body parts, and indeed, to inspire the concept of unconstrained counting in a way the more limited capacity of the body might not. And once they have been made, marks have the potential for being used as a reference, and both behaviors and the stimuli produced have the potential for being named: to tally, the count, thirteen. But tallies don’t mean there had to be a word for each quantity represented by the marks, since they can be used without lexical numbers like the rosary is: One bead means one prayer, and all the beads mean enough prayers were said, not that someone prayed fifty times. Tallies also don’t exclude the possibility of an associated lexicon; it’s just something we can’t assume, either way.

EVIDENCE OF TALLIES IN THE ANCIENT NEAR EAST Archaeological evidence of early tallies in the Southern Mesopotamian Plain where writing would later develop is rare. This is because the region was mostly uninhabited until around 9000 BCE; between 9000 and 6000 BCE, settlements were small and isolated, making it more difficult to discover sites and artifacts.40 Artifacts made of organic materials like wood and bone tend to degrade in the region’s climate, making it even less likely they will be found. These deposition and preservation factors mean that few artifacts made of organic materials are found at sites located inland, far from the Mediterranean coast,41 even though bone and wood were common materials. If rare, artifacts made of organic materials have nonetheless been found in the Levant, the Mediterranean coastal regions where wild grains were collected and consumed around 20,000 years ago and cultivation is thought to have started around 10,000 years ago.42 These artifacts are pieces of bones with markings on them, and some look like they possibly might have been tallies. The oldest is an incised bone found at Kebara Cave, modern Israel, a site dated to 60,000–48,000 years ago, the Late Middle Paleolithic; incised bones have also been found at nearby Ha-Yonim Cave, a site dated to 30,000–12,000 years ago, the Late Upper Paleolithic.43 Nissen, The Early History of the Ancient Near East, 9000–2000 B.C. Coinman, ‘Worked Bone in the Levantine Upper Paleolithic: Rare Examples from the Wadi Al-Hasa, West-Central Jordan’, pp. 114–115. 42 Hours et al., Atlas des Sites du Proche-Orient (14,000–5,700 BP); Nissen, The Early History of the Ancient Near East, 9000–2000 B.C.; Stordeur and Abbès, ‘Du PPNA au PPNB: Mise en Lumière d’une Phase de Transition à Jerf El Ahmar (Syrie)’. 43 Davis, ‘Incised Bones from the Mousterian of Kebara Cave (Mount Carmel) and the Aurignacian of Ha-Yonim Cave (Western Galilee), Israel’. 40 41

8. FINGERS AND TALLIES Period Late Middle Paleolithic

Late Upper Paleolithic

Time Site 60,000–48,000 Kebara Cave, years ago Israel Ha-Yonim Cave, 30,000–12,000 Israel years ago Ksar’Aqil, Lebanon

1 5 1

25,000–19,000 Ain el-Buhira, years ago Jordan

2

17,000–15,000 Jita, Lebanon years ago

3

10,500 years ago

1

Epipaleolithic 9000–7000 BCE

8300–5500 BCE

7500–7000 BCE

Neolithic

Bones

7250–6750 BCE

7000–3300 BCE

Öküzini Cave, Turkey Tell Kurdu, Turkey Tell Turmus, Israel Jericho, Israel Girikihaciyan, Turkey Sakçe Gözü, Turkey Tell Arpachiyah, Iraq Yarim Tepe II, Iraq Cayönü Tepesi, Turkey Tell al-Judaidah, Turkey Byblos, Lebanon

6000–5500 BCE

Yarim Tepe I, Iraq

5900–5000

Hajji Firuz Tepe, Iran

BCE

2 1

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References Davis, ‘Incised Bones from the Mousterian of Kebara Cave (Mount Carmel) and the Aurignacian of Ha-Yonim Cave (Western Galilee), Israel’. Tixier, ‘Poinçon décoré du paléolithique supérieur à Ksar’Aqil (Liban)’. Coinman, ‘Worked Bone in the Levantine Upper Paleolithic: Rare Examples from the Wadi Al-Hasa, West-Central Jordan’. Copeland and Hours, ‘Engraved and Plain Bone Tools from Jita (Lebanon) and Their Early Kebaran Context’. Yalçinkaya et al., ‘Les Occupations Tardiglaciaires du site d’Öküzini (Sud-Ouest de la Turquie). Reese, ‘On the Incised Cattle Scapulae from the East Mediterranean and Near East’. Dayan, ‘Tell Turmus in the Ḥuleh Valley’.

Kenyon and Holland, Excavations at Jericho V; Marshall, ‘Jericho Bone Tools and Objects’. Watson and LeBlanc, Girikihaciyan, A Halafi8 an Site in Southeastern Turkey. Du Plat Taylor, Seton Williams, and Waecht2 er, ‘The Excavations at Sakçe Gözü’. Mallowan and Cruickshank Rose, ‘Excavamany tions at Tall Arpachiyah, 1933’. Merpert et al., ‘The Investigations of the 1 Soviet Expedition in Iraq 1973’. Redman, ‘Early Village Technology: A View 2 through the Microscope’. Braidwood et al., Excavations in the Plain of 2 Antioch, Vol. 1. Von den Driesch and Boessneck, ‘Osteologische Besonderheiten vom Morro 2 de Mezquitillá Malága’. Merpert and Munchajev, ‘Excavations at Yarim Tepe 1970’; Munchajev and Merpert, 4 ‘Excavations at Yarim Tepe 1972’. Voigt, Hajji Firuz Tepe, Iran: The Neolithic 1 Settlement. 1

Table 8.1. Possible early tallies. Worked bones possibly used as tallies have been found at multiple sites dated to the Paleolithic, Epipaleolithic, and Neolithic periods. Data on the earliest worked bones possibly used as tallies were compiled from the sources indicated; also see Schmandt-Besserat, Before Writing: From Counting to Cuneiform, Vol. 1.

Similar artifacts have been discovered at Ksar’Aqil44 (Late Upper Paleolithic) and Jita45 (17,000–15,000 years ago) in Lebanon and Ain el-Buhira46 (25,000–19,000 years ago) in Jordan. Incised bones dated to the Epipaleolithic (12,000 years ago to 8300 BCE) and Neolithic (8300–4500 BCE), generally cattle scapulae, have also been found at sites in Tixier, ‘Poinçon Décoré du Paléolithique Supérieur à Ksar’Aqil (Liban)’. Copeland and Hours, ‘Engraved and Plain Bone Tools from Jita (Lebanon) and Their Early Kebaran Context’. 46 Coinman, ‘Worked Bone in the Levantine Upper Paleolithic’. 44 45

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Israel, Palestine, Lebanon, Turkey, Iran, and Iraq (Table 8.1 and Fig. 8.1). These artifacts have been ascribed a variety of possible purposes and functions, including counting, making music, and being tools like awls or hunting implements.

Fig. 8.1. Geographic distribution of the oldest sites with possible tallies. Worked bones possibly used as tallies and dated to the Paleolithic, Epipaleolithic, and Neolithic periods have been found at sites in the Levant and Zagros Mountains. Adapted from an image in the public domain.

Although the worked bones from Ha-Yonim Cave (Fig. 8.2, bottom) and Ksar’Aqil (Fig. 8.3) might look more like tallies,47 those from Kebara, Jita, and Ain el-Buhira (Fig. 8.2, top and Fig. 8.4) appear more decorative in their purpose. In fact, the ones from Jita and Ain el-Buhira are thought to have been tools: respectively, an awl and a hunting implement.48 The artifacts listed in Table 8.1 have been subjected to descriptive analyses, but none has been conclusively identified as a tally. For example, marks of a ‘parallel and repeated nature’ on two worked bones from Cayönü Tepesi, a site in southeastern Turkey dated to 7250–6750 BCE, suggested they might have been used for counting; however, the 13 marks on one of the two were not mathematically suggestive, and the wear on both was also consistent with several alternative uses—calendars, music making, and even sharpening other tools.49 Certainly, regularity attests to the intentionally of the marks; the complexity of many of the artifacts shows that time and Schmandt-Besserat, Before Writing: From Counting to Cuneiform, Vol. 1, p. 159. Coinman, ‘Worked Bone in the Levantine Upper Paleolithic’, p. 118; Copeland and Hours, ‘Engraved and Plain Bone Tools from Jita (Lebanon) and Their Early Kebaran Context’, p. 297. 49 Redman, ‘Early Village Technology: A View through the Microscope’, p. 258; also see Figs 6.4 and 6.5 on p. 260. 47 48

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effort were invested in their manufacture, as well as deliberation and skill. The only question really left unanswered is whether these capacities and capabilities were also focused on representing numbers. If numbers emerged as early as the other evidence suggests, there’s no reason they couldn’t have been. It’s also worth remembering that making marks represents cognitive opportunities, as discussed earlier, and that sooner or later, the ability to make marks intentionally is likely to be turned to numerical purposes.

Fig. 8.2. Worked bones from Kebara (top) and Ha-Yonim (bottom), Israel. While the function of these artifacts is unknown, ‘it might be supposed that they served as some method of keeping quantitative records’,50 though the possibility they were used for other purposes cannot be excluded. The images were originally published in Davis, ‘Incised Bones from the Mousterian of Kebara Cave (Mount Carmel) and the Aurignacian of Ha-Yonim Cave (Western Galilee), Israel’, Paléorient 2(1), p. 182, Figs 1 and 3. They are republished with the kind permission of Paléorient.

50 Davis, ‘Incised Bones from the Mousterian of Kebara Cave (Mount Carmel) and the Aurignacian of Ha-Yonim Cave (Western Galilee), Israel’, p. 181.

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Fig. 8.3. Worked bone from Ksar’Aqil, Lebanon. (Top) The artifact was found at the Late Upper Paleolithic site of Ksar’Aqil by French archaeologist Jacques Tixier in the early 1970s. (Bottom) Tixier drew and numbered the five rows of marks. While most are single lines, a few are X- or V-shaped. While all five rows contain between 32 and 35 marks, in several cases, smaller marks are interspersed between those of the more typical larger size, a disposition suggesting their totality may have been unplanned. The rows are arranged along the length of the artifact with a fair degree of regularity, and the marks are roughly parallel, evenly spaced, and similarly sized. These characteristics meet several of the criteria for determining a quantificational intent. Tixier cautioned he could not say for certain whether the markings served any purpose other than decoration,51 and in fact, he thought the artifact may have been an awl. The images were originally published in Tixier, ‘Poinçon Décoré du Paléolithique Supérieur à Ksar’Aqil (Liban)’, Paléorient 2(1), p. 190, Figs 3 and 4. They are republished with the kind permission of Paléorient.

51

Tixier, ‘Poinçon Décoré du Paléolithique Supérieur à Ksar’Aqil (Liban)’, p. 192.

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Fig. 8.4. Worked bones from Jita, Lebanon and Ain el-Buhira, Jordan. Both devices appear more decorative than quantificational in intent. (Top) The Jita artifact was identified as an awl.52 (Bottom) The Ain el-Buhira artifact may have been ‘an elaborately notched projectile representing some type of hunting implement’.53 Only a portion of it is shown. The image of the artifact from Jita was originally published in Copeland and Hours, ‘Engraved and Plain Bone Tools from Jita (Lebanon) and Their Early Kebaran Context’, Proceedings of the Prehistoric Society 43, p. 296, Fig. I.3. It is reproduced with the kind permission of Cambridge University Press. The image of the Ain el-Buhira artifact was originally published in Coinman, ‘Worked Bone in the Levantine Upper Paleolithic: Rare Examples from the Wadi Al-Hasa, West-Central Jordan’, Paléorient 22(2), p. 117, Fig. 4. It is republished with the kind permission of Paléorient.

By the Chalcolithic (4500–3300 BCE) and Early Bronze Age (3300–2000 BCE), the central portion of the Mesopotamian Plain had been settled and was inhabited by populations of increasing size. During these periods, artifacts and tallies are found archaeologically, or they are implied by aspects of early accounting, or they are mentioned in early texts. Incised bones dated to 4500–2700 BCE have been found at multiple sites in Lebanon and Syria,54 though again, none have been conclusively identified as tallies. Bookkeeping in the 4th and 3rd millennia BCE likely included ‘a number of devices such as tally sticks and tokens’,55 an assessment based on practical considerations and the sums impressed on clay artifacts, recording and calculating that would have been difficult to accomplish without the use of such artifacts. Tallies are mentioned explicitly in a text from the 2nd millennium BCE, ‘The debate between grain and sheep’. Lines 130– 133 have been translated as, ‘Every night your count is made and your tally-stick put into the ground, so your herdsman can tell people how many ewes there are and how 52 Copeland and Hours, ‘Engraved and Plain Bone Tools from Jita (Lebanon) and Their Early Kebaran Context’, p. 297. 53 Coinman, ‘Worked Bone in the Levantine Upper Paleolithic’, p. 118. 54 Reese, ‘On the Incised Cattle Scapulae from the East Mediterranean and Near East’. 55 Englund, ‘Grain Accounting Practices in Archaic Mesopotamia. Changing Views on Ancient Near Eastern Mathematics’, p. 23.

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many young lambs, and how many goats and how many young kids’.56 And tallies made of wood dated to the middle and late 1st millennium BCE have been found; they are unambiguous regarding their purpose, something established through other records and inscriptions found with them.57

THE CONTEXT FOR EVALUATING ARTIFACTS FOR USE AS TALLIES If the artifacts in Table 8.1 had been made within the past few hundred years, we still wouldn’t know what they were used for just by looking at them, but we might have access to historical observations of people making and using such devices. But even without observing, we wouldn’t automatically eliminate the possibility the artifacts were tallies, since we assume numbers are well within the capacity of any modern people. Of course, this assumption becomes less tenable when we start going backward in time. Our evolutionary lineage includes species of bipedal apes like H. habilis, which lived 2.3 to 1.4 million years ago, and H. erectus, which overlapped it at 2 million to 140,000 years ago. Both differed from us anatomically and behaviorally, with smaller, more elongated primate brains and behaviors that fall somewhere between those of apes and humans, as judged from the remains of their material culture. Our own species, H. sapiens, emerged roughly 400,000 years ago. Their earliest incarnations resembled us in terms of their brain sizes and shapes, but they still didn’t act like we do. Only within the last 200,000 to 100,000 years did our ancestors finally begin leaving material traces of behaviors unambiguously recognizable as things we do, like creating and using tools for hunting and stringing and wearing necklaces of shell beads. We assume H. sapiens groups within the last 40,000 years were a lot like us; they may have lacked the complex accumulated knowledge and material culture we would eventually inherit from them, but they were certainly well started on the process of developing it. For them, we assume a competence analogous to our own. This assumption is thought especially true for people living in the past 10,000 years, when the pace of the invention, elaboration, and accumulation of human material culture escalated dramatically (Fig. 8.5). When we evaluate whether it’s possible the devices found in the pre-Neolithic Levant and Neolithic Mesopotamia were used for counting, there are several things to keep in mind. First, as I explained at the beginning of the chapter, what we find does not represent the entire class of possible one-dimensional devices used for accumulation and recording cross-culturally. We find only a small subset: those devices made of a material durable enough to preserve, that we were also lucky enough to find, and that also happen to look like tallies. As a result, the archaeological record underestimates the one-dimensional counting artifacts, devices like tallies that accumulate but don’t group quantity.

56 Electronic Text Corpus of Sumerian Literature, ‘Translation of “The Debate between Grain and Sheep”’. 57 Henkelman and Folmer, ‘Your Tally Is Full! On Wooden Credit Records in and after the Achaemenid Empire’, pp. 138–141.

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Fig. 8.5. Pace of technological invention, elaboration, and accumulation. As can be seen, the pace of developing the technology that characterizes us today escalated dramatically at the beginning of the Neolithic, roughly 10,000 years ago. The scale is logarithmic, rather than linear. On a linear scale, shown in the insert, the change in pace is even more impressive: It’s basically flat until the 10,000-year mark, when it abruptly turns 90° and skyrockets vertically. Tally sticks, highlighted in larger font, are estimated to have started between 100,000 and 10,000 years ago with devices like the Lebombo and Ishango bones; they are consistent with the other technological developments of the period, including bone tools, bows, and cave painting. Image created by the author.

Second, one-dimensional devices like tallies represent an important transition between counting with the body and counting with material structures, something we can observe and verify cross-culturally. Of course, material devices aren’t mandatory for counting to numbers higher than what the fingers can afford: A few dozen societies in Papua New Guinea count to higher numbers using their bodies as a tally.58 However, while body-counting is certainly an effective method of counting, it doesn’t have the same potential for elaborating, accumulating, and distributing knowledge that materiality does. And the people of the Late Upper Paleolithic and Neolithic periods were using materiality to solve problems and accumulate knowledge to an unparalleled extent; it would be unusual and strange indeed if their numbers were exempt from this. 58 Lean, ‘Counting Systems of Papua New Guinea and Oceania’; Saxe, Cultural Development of Mathematical Ideas.

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Another point to keep in mind is that the two-dimensional complexity of the numerical tokens, as we’ll see in Chapter 9, implies there would have been at least one simpler, one-dimensional precursor not part of the body: some kind of onedimensional device besides the fingers. This is because inventing a two-dimensional method like tokens directly from finger-counting would entail a significant technological leap, one large enough to be less likely. That something one-dimensional and not the fingers bridged the gap is also supported by differences in the organization of the tokens, which were based on cycles of 10 and 6, and Mesopotamian finger-counting, which patterned number words and notations with cycles of 5, 10, and 20. The probability that tallies or similar material technologies bridged the gap also must be viewed in light of the claim that Mesopotamian numbers were invented during the Neolithic. This means numbers would have gone from scratch—the perceptual experience of quantity—directly to tokens, an even larger and more implausible leap than going from finger-counting to tokens. It’s not parsimonious because it assumes Neolithic peoples wouldn’t have had numbers at a point when their demographic and technological factors indicate they should have had them, and that they then would have become numerically inventive to a degree not seen in any other people. Such assumptions aren’t realistic: Invention doesn’t generally span large technological chasms; rather, societies tend to build incrementally on what they’ve inherited. In numbers, this means societies, more often than not, supplement finger-counting with a variety of one-dimensional devices. The ethnographic record is not merely full of examples of this, the condition of counting with fingers, language, and one-dimensional devices is also relatively common. Societies before the middle of the Neolithic were unlikely to have been numerically incompetent, since they too were highly inventive. In the Late Upper Paleolithic and early Neolithic, the dispersed, semi-nomadic peoples who had historically subsisted through hunting and gathering were entering into new relations with resources and landscapes: They were cultivating cereal, domesticating animals, and becoming sedentary, which required them to build and maintain shelters.59 By the 8th millennium BCE, settlements like Tell Aswad, Tel Mureybet, and Cheikh Hassan in Syria and Ganj Dareh Tepe and Tepe Asiab in Iran were noticeably larger in population, something that increases requirements for producing and storing food,60 which in turn is just the very kind of thing that motivates numerical elaboration.61 Early Neolithic peoples were also coming into contact with each other and exchanging information like never before, as attested by the way the new technologies spread throughout the region. There is also evidence of long-distance trade, as highly valued materials like obsidian have 59 Braidwood et al., Prehistoric Archaeology along the Zagros Flanks; Childe, Man Makes Himself; Hesse, ‘Cultural Interaction and Cognitive Expressions in the Formation of Ancient Near Eastern Societies’; Lloyd, The Archaeology of Mesopotamia; Pollock, Ancient Mesopotamia: The Eden That Never Was; Steiner and Killebrew, The Oxford Handbook of the Archaeology of the Levant: C. 8000–332 BCE. 60 Schmandt-Besserat, ‘The Emergence of Recording’. 61 Divale, ‘Climatic Instability, Food Storage, and the Development of Numerical Counting: A Cross-Cultural Study’.

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been found 300 kilometers from the nearest deposits.62 Along with the other technologies, numbers would have spread as well, as they are a linguistic domain known to traverse languages and cultures with an unusual ease and speed. Contact would have provided a mechanism for their elaboration as well, since exposure to number systems with different characteristics would have provided opportunities for people to notice differences, a viable mechanism for explicating and exploiting the properties in question. A final issue with the idea there were tallies in the Levant during the Late Upper Paleolithic and in the Mesopotamian Plain during the Neolithic and later periods is the cultural continuity between the two. I do not suggest there is an unbroken lineage. There is, however, genetic evidence connecting the earlier Levantine and later Mesopotamian populations.63 Their languages were most certainly related: Akkadian, an East Semitic language spoken in Mesopotamia, was related to West Semitic languages spoken in the Levant, like Aramaic. Technological similarity connects the earlier cereal cultivation of the Levant to the later industrialized agriculture of Mesopotamia, and reason notes that without the former, the latter wouldn’t have emerged. While such links are tenuous rather than dispositive and may leave the historian dissatisfied, linguistic change, ‘gene flow, and cultural borrowing can obviously operate in almost all human contexts, except perhaps in situations of extreme isolation’.64 Views of these cultures as independent, non-overlapping, and self-contained is being displaced by more nuanced perspectives that recognize the Late Upper Paleolithic Levant and Neolithic Mesopotamia as characterized by vibrant, expanding cultural networks and increasing interactions on local and regional geographic scales, as well as across time. 65 But beyond all of this, my foundational assumption is that the Ancient Near Eastern peoples would have been like any other with respect to inventing and using their numbers. This includes the kind of devices they were likely to have used, as well as the demographic and technological conditions and contexts under which they were most likely to have used them. It also sets the stage for the next technology that emerged for counting: the Neolithic tokens justly famous as the precursors of written notations for numbers.

Schmandt-Besserat, ‘The Emergence of Recording’, p. 874. Lazaridis et al., ‘Genomic Insights into the Origin of Farming in the Ancient Near East’. 64 Bellwood, ‘Early Agriculturalist Population Diasporas? Farming, Languages, and Genes’, p. 185. 65 Goring-Morris and Belfer-Cohen, ‘Neolithization Processes in the Levant: The Outer Envelope’; Hesse, ‘Cultural Interaction and Cognitive Expressions in the Formation of Ancient Near Eastern Societies’; Watkins, ‘Supra-Regional Networks in the Neolithic of Southwest Asia’. 62 63

CHAPTER 9. THE NEOLITHIC CLAY TOKENS Discussions of the Neolithic tokens—small, geometrically shaped objects, typically made of clay, thought to have been used as numerical counters—do not begin with the tokens themselves, but rather, with token-like impressions on clay artifacts dated to the mid-to-late 4th millennium BCE. It’s the correspondences of shape, size, and quantity between the tokens and impressions, and between the impressions and later numerical notations, that attest to the numerical meaning and likely physical organization of the tokens (Fig. 9.1). This is significant in light of two issues previously mentioned. First, numerical meaning and use cannot be established with similar certainty for precursor technologies like tallies, as these things cannot be ascertained solely from the artifacts themselves. Second, a timeline for when precursor technologies like fingers might have emerged cannot be established except in the relative sense of likely to have been first, since they are attested by linguistic evidence and implied neurologically rather than consisting of datable materials. Tokens, in contrast, are unambiguously numerical and can be pegged to an absolute chronology. They inject a couple of issues into the discussion, and occasion new attributes for the concept of number, the topic of this chapter. Archaeologists have long been excavating ‘small and seemingly insignificant clay objects of various forms and sizes … from early periods all over the Middle East’. 1 Several have speculated these artifacts may have been used as numerical ‘tokens’ or ‘counters’: Vivian Broman,2 A. Leo Oppenheim,3 and Pierre Amiet. The latter described several bullae found at Susa, an Elamite city east of the Tigris River in the lower Zagros Mountains. Bullae were hollow spheres of clay used to contain smaller clay artifacts shaped into cones, tetrahedrons, spheres, and disks; age-wise, they were roughly contemporary to early Uruk (3500–3200 BCE), the period of early Mesopotamian history named for the city that typified its culture.

Friberg, ‘Preliterate Counting and Accounting in the Middle East: A Constructively Critical Review of Schmandt-Besserat’s Before Writing’, p. 478. 2 Broman, Jarmo Figurines. 3 Oppenheim, ‘On an Operational Device in Mesopotamian Bureaucracy’. 1

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Fig. 9.1. Correspondences between plain tokens, proto-cuneiform notations, and cuneiform number signs. Plain tokens (top) were used for accounting before writing was invented. Numerical impressions (not shown, but see Fig. 9.2, right) and proto-cuneiform numerical signs (middle) emerged during the early Uruk periods (3500–3200 BCE), cuneiform number signs (bottom) as early as 3000 BCE. Correspondences of shapes, sizes, quantities, and grouping relations between cuneiform numbers, proto-cuneiform numerical signs, numerical impressions, and tokens attest to the numerical meaning of the tokens, at least at the point in time where the tokens and impressions coincide in the mid-to-late 4th millennium BCE. Notice the different technologies trend toward higher numbers, with tokens expressing the lowest and cuneiform signs the highest. Tokens thought to have represented amounts in the hundreds have been dated to the Pre-Uruk V period (8500– 3500 BCE),4 a quality implying the number system was already relatively mature by the Neolithic. Signs for ‘3600’ (602) and ‘216,000’ (603) are attested as early as 3000 BCE,5 showing an expansion of the highest number expressible in numerical signs from hundreds to hundreds of thousands between the late Neolithic and Early Bronze Age. The plain tokens were adapted from Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’, p. 29, Fig. 24. The proto-cuneiform notations were redrawn from Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East, pp. 28–29, Fig. 28. The cuneiform number signs are Unicode characters.

Amiet believed the enclosed tokens represented a numerical series, with the substitution of a tetrahedron for a cone perhaps differentiating the type of commodity being

4 5

Cuneiform Digital Library. University of Pennsylvania, The Pennsylvania Sumerian Dictionary.

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enumerated.6 He also noted bullae appeared to precede the appearance of impressed clay tablets in the archaeological record, a discovery at odds with his apparent expectation that tablets should logically precede bullae (Fig. 9.2).

Fig. 9.2. Clay bulla (W 20987,08) and early numerical impressions (OIM A21310). (Left) The clay bulla with impressions and tokens comes from the ancient Sumerian city of Uruk and has been dated to the Pre-Uruk V period. It is held in the collection of the German Archaeological Institute, Berlin, where it is on loan from the University of Heidelberg. It exemplifies a technology that emerged in the mid-to-late 4th millennium BCE. Its surface is marked with seven impressions that correspond well to the shapes, sizes, and quantities of the associated tokens: five tetrahedra and two disks.7 (Right) The numerical tablet comes from Tutub, a city located northeast of Sumer proper. It has been assigned to the Uruk V period (3500–3350 BCE) and is held in the collection of the Oriental Institute of Chicago. Its obverse face is impressed with shapes analogous to one spherical and five cylindrical tokens.8 Image of clay bulla © DAI/Orient-Abteilung and reproduced with permission from the German Archaeological Institute, Berlin. Image of early numerical impressions courtesy of the Oriental Institute of the University of Chicago.

Perhaps the scholar most widely associated with the tokens is archaeologist Denise Schmandt-Besserat. She extensively developed the idea the tokens were numerical counters, explaining the archaeological sequence of technology as follows (Fig. 9.3): Tokens were first. Because they were loose, they were at some point contained in un6

Amiet, Mémoires de la Délégation Archéologique en Iran, Tome XLIII, Mission de Susiane, Vol. I,

p. 69. 7 Damerow and Meinzer, ‘Computertomografische Untersuchung Ungeöffneter Archaischer Tonkugeln aus Uruk, W 20987,9, W 20987,11 und W 20987,12’. 8 Delougaz, Hill, and Lloyd, Private Houses and Graves in the Diyala Region; Frankfort, Progress of the Work of the Oriental Institute in Iraq, 1934/35: Fifth Preliminary Report of the Iraq Expedition, p. 25, Fig. 19.

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marked bullae, and subsequently by bullae onto whose surfaces they were impressed before their enclosure. The dual method—impressions outside, tokens inside— prompted the realization that two forms were unnecessary, whereupon tokens were discarded as representations of quantity and bullae flattened out to become tablets bearing token-like impressions. She noted the number of token finds sharply decreased in conjunction with this technological sequence, an effect she attributed to tokens being replaced by notations.

Fig. 9.3. Chronology of artifacts used in Mesopotamian accounting and mathematics. Possible plain tokens have been dated as early as the 10th millennium BCE9 and as late as the 1st millennium CE.10 This expands the time frame of Schmandt-Besserat’s original dataset, which spanned the 9th through the 1st millennia BCE. The cuneiform script would continue into the first century of the Common Era; its last known use dates to 75 CE.11 Imaged based on Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’, p. 16, Fig. 9; also see Englund, ‘Texts from the Late Uruk Period’; MacGinnis et al., ‘Artefacts of Cognition: The Use of Clay Tokens in a Neo-Assyrian Provincial Administration’; Schmandt-Besserat, Before Writing: From Counting to Cuneiform.

Schmandt-Besserat comprehensively catalogued the artifacts thought to be tokens as these were known up to the early 1990s, over 8,000 artifacts spanning the entire Mesopotamian region and dated to between the 9th and 1st millennia BCE. Each was placed into one of 16 different categories or types, often differentiated by subtype, in a typology she created for this purpose. She published her findings as a two-volume set of analysis and data.12 Schmandt-Besserat also differentiated plain tokens, simple geometric forms whose numerical meaning was attested by their correspondences with later, unMoore and Tangye, ‘Stone and Other Artifacts’. MacGinnis et al., ‘Artefacts of Cognition: The Use of Clay Tokens in a Neo-Assyrian Provincial Administration’. 11 Geller, ‘The Last Wedge’. 12 Schmandt-Besserat, Before Writing: From Counting to Cuneiform. 9

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ambiguous numerical signs, from complex tokens, ones modified by incision, punctation, or other techniques in ways she believed were intended to convey systematic linguistic meaning. Simply, Schmandt-Besserat proposed that ‘“plain” tokens were forerunners of impressed protoliterate number signs, while “complex” tokens were forerunners of incised protocuneiform signs of writing’.13

TOKENS AND NUMERICAL MEANING The so-called plain tokens took the shapes of spheres, cones, disks, cylinders, and tetrahedra. They tended to occur in two sizes, small and large, though occasionally an intermediate size was noted; size was a component of representing numerical value. Their quantities suggest numerical relations called bundling, in which a token of one shape and size was equal to six or ten tokens of another shape or size. Specific combinations of shapes, sizes, and bundling relations encoded the type of commodity being enumerated. In the proto-cuneiform notations that closely resembled the earlier tokens, there were about a dozen systems for counting different types of objects, differentiated by the combinations of shapes, sizes, and bundling relations.14 Some of the proto-cuneiform counting systems were clearly derived from others. Derived systems were differentiated from basic systems by superimposed markings, usually horizontal or vertical lines. For example, damaged goods were indicated by marking numerical signs with a long horizontal line, reminiscent of the way we might cross out a number to indicate its loss of importance. Some proto-cuneiform notations appeared in more than one counting system, where they may have had a different numerical value. This made their meaning context-dependent, a multiplicity of potential value known as polyvalence. The distribution of numerical meaning across systems and signs has helped foster the perception the associated numerical concepts were concrete rather than abstract. Many of the traits of the proto-cuneiform numerical signs—multiple systems, derived and basic systems, modification of derived signs, relational meaning, contextdependent meaning, and polyvalence—appear to have characterized the earlier tokens as well. And as many as ten of the proto-cuneiform counting systems may have been used with tokens before writing was invented, as suggested by the signs on impressed artifacts with token assemblages, as well as numerical signs of the Uruk V period (Table 9.1).

Friberg, ‘Preliterate Counting and Accounting in the Middle East’, p. 484. Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. 13 14

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System Sexagesimal System S Sexagesimal System S' Bisexagesimal System B U4 System U GAN2 System G DUG Systems Db and Dc

Use in Counting Most discrete objects Certain objects: dead animals, liquid products Food rations: grain products, cheese, fresh fish Calendar units Area measures Milk products like dairy fats

ŠE System Š ŠE Systems Š' and Š''

Grain: barley Grain: malt, emmer

Attested By Both Uruk V impressions Both Both Both Impressed bullae with tokens Uruk V impressions Impressed bullae with tokens

Table 9.1. Counting systems possibly used with tokens. Analysis of numerical signs from impressed artifacts with associated token assemblages (see Table A.1) and numerical impressions of the Uruk V period (Table A.2) suggests as many as ten counting systems, the majority of those associated with proto-cuneiform numerical signs, may have been used in token-based accounting before the invention of writing. System names and qualifying signs follow the typology and conventions established by Nissen and his colleagues in their 1993 publication.

The idea the plain tokens had numerical meaning—at least in the mid-to-late 4th millennium BCE, when this meaning is attested by impressed bullae and tablets and correspondences with later numerical notations—has been so well accepted, it has achieved the status of fact: ‘There is no reason to doubt that the (mostly) plain tokens enclosed in spherical envelopes belonged to a small number of preliterate systems of “number tokens”, very much similar to the now [well-known] protoliterate systems of impressed number notations’.15 The numerical meaning of tokens from the Pre-Uruk V period can be verified by comparing the known token assemblages to the notations on the associated bullae; the two correspond closely (Table A.1). Also widely accepted are Schmandt-Besserat’s explanation of the technological sequence—tokens, bullae as unmarked containers, impressed bullae, tablets, and writing—as well as her general view of tokens as important precursors of writing, if not always within Assyriology, at least in the popular imagination. Despite widespread acceptance of the idea the plain tokens had numerical meaning in the mid-to-late 4th millennium BCE, Schmandt-Besserat’s conclusions have not been accepted in toto. Archaeologist Hans Nissen, psychologist Peter Damerow, and Assyriologist Robert Englund analyzed and codified the numerical relations between the numerical signs and systems of proto-cuneiform writing, relations the 4thmillennium plain tokens exemplified as well (Table A.1). This contradicted SchmandtBesserat’s belief the tokens had no inherent numerical meaning or relation to one another beyond representing, in a one-to-one fashion, whatever it was they counted. In the one-to-one relation, each item in a set is matched to an item in a second set, with none left unmatched or matched more than once. Within each set, the items have the same value, since being different in value either disqualifies them from being members 15

Friberg, ‘Preliterate Counting and Accounting in the Middle East’, p. 492.

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of the set in the first place or limits how the items are matched across sets. In comparison, differences of value are implicit to relations like the bundling that characterized the tokens and subsequent technologies, and they preclude the relations from being one-to-one. Viewing tokens as related one-to-one despite the differences of value encoded by their shapes and sizes may have comfortably fit with their being conceptualized as the first Mesopotamian counting technology and the numbers they represented as concrete, rudimentary, and used in the absence of concepts or words. However, the numerical relations between tokens demonstrated by Nissen and his colleagues and apparent in analyses of token assemblages and early notations (Tables A.1 and A.2) exclude the possibility tokens were used in one-to-one fashion. Bundling or grouping also constitutes a second dimension of numerical representation with implications of its own. Its complexity makes the idea the tokens were the first technology used for counting highly improbable, a conclusion supported by ethnographic observations suggesting one-dimensional devices are more common, especially with emerging numbers. Bundling thus implies the tokens would have had one or more one-dimensional precursors, something Schmandt-Besserat apparently recognized,16 but whose implications she didn’t pursue. Bundling also excludes the possibility the tokens were used without number concepts or words, since concepts and words are implicit to the operations that exchange token values—bundling and debundling. In the exchange operation of bundling, the sixth or tenth token of a lower value, for example, would trigger an exchange with the single token representing the equivalent higher value; conversely, in debundling, one single higher-value token would be exchanged for the equivalent number of lower-value tokens. Implicit to such operations is the ability to understand and manipulate non-subitizable quantities like six and ten, and this makes their non-availability as words or concepts implausible, if not impossible. The idea numerical tokens were used throughout the entire span of time claimed by Schmandt-Besserat has been rejected. Even in the mid-to-late 4th millennium BCE, when their numerical meaning was most assured, token shapes, sizes, and quantities varied both ‘temporally and regionally’.17 This variability increases the further back in time we look, so the idea there was a single coherent system of numerical tokens beginning as early as the 9th or 10th millennia BCE is untenable. There are other issues perplexing the identification of small clay objects as tokens, detailed in the next section. Cumulatively, these issues make it unlikely that all of the items listed in SchmandtBesserat’s catalogue, perhaps even many of them, were numerical counters. This in turn weakens the claim that the use of tokens as numerical counters may have emerged as early as the 9th millennium BCE. Other estimates have placed the emergence around 6000 BCE, consistent with the movement of people into southern Mesopotamia, the growth of settlement and population during the early Neolithic18 and the emergence of technologies like seals.

Schmandt-Besserat, Before Writing, Vol. 1, pp. 185–189. Friberg, ‘Preliterate Counting and Accounting in the Middle East’, p. 484. 18 Robson, Mathematics in Ancient Iraq: A Social History. 16 17

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Arguably, the most contested aspect of Schmandt-Besserat’s work has been her idea the complex tokens were a notational system that directly prefigured writing.19 Some of the modifications to plain tokens she had noted ‘may possibly have been the precursors of, or at least inspired, the introduction of signs belonging to several known protoliterate “derived” systems of number notations’.20 That is, rather than denoting language and prefiguring written signs per se, the markings may have helped differentiate derived counting systems in a fashion similar to conventions used in the later proto-cuneiform numerical signs. And taken as a possible complement of conventionalized signs, complex tokens correspond only minimally to the pictures and conventionalized signs used as the first commodity labels, especially once tokens with horizontal or vertical lines similar to those marking derived counting signs in proto-cuneiform have been subtracted. There are other grounds for rejecting the complex-token hypothesis: They may not have been the earliest form of representing meaning, since their terminus post quem, the earliest time they might have emerged, is not certain to have been the 9th millennium BCE, and cylinder seals—engraved objects, usually stone, carved with meaningful designs and motifs and designed to be rolled onto clay surfaces to leave impressions— have been dated to the 6th millennium BCE.21 Archaeologically, there are discontinuities between the prevalence of goods and the complex tokens purported to represent them: Complex tokens bearing the quartered circle that signified sheep are rare, yet sheep were a common commodity.22 There are also neurological and biomechanical objections to the idea complex tokens directly prefigured writing,23 as I’ll explain in Chapter 10. Undoubtedly, complex tokens were meaningful to those using them, and their inclusion in token sets may have indicated things like the type of commodity or the origin or ownership of the goods, codes understood between trading partners. Certainly, they were part of a general technological sequence that cumulatively set the conditions out of which writing developed. This role, however, is far less crucial to the invention of writing than the one Schmandt-Besserat envisioned for them.

ISSUES IN INTERPRETING TOKENS AS NUMERICAL COUNTERS What tokens might reveal about Mesopotamian numbers before they were impressed into clay is challenged by several archaeological, legal, and financial issues limiting their interpretation as a technology for representing and manipulating quantity: x

Any small object made of clay tends to be classified as a numerical token, whether or not they were one. There are no standard criteria for distinguish-

19 Englund, ‘Review: Denise Schmandt-Besserat, How Writing Came About’; Friberg, ‘Preliterate Counting and Accounting in the Middle East’; Zimansky, ‘Review of Denise Schmandt-Besserat’s Before Writing, Volumes I and II’. 20 Friberg, ‘Preliterate Counting and Accounting in the Middle East’, p. 485. 21 Shendge, ‘The Use of Seals and the Invention of Writing’. 22 Zimansky, ‘Review of Denise Schmandt-Besserat’s Before Writing, Volumes I and II’. 23 Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’.

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ing numerical tokens from small clay objects used for any other purpose, and clay was a common manufacturing material used throughout the Mesopotamian region and across its history for a wide variety of functions. There is also no single typological system for tokens thought to have been used for counting at either the type or the subtype level, making it difficult to compare the token finds published by different authors. x

Whatever their use, small objects made of clay have historically been undercollected during excavation or under-documented in publication or both.24 The data are thus incomplete, which means their analyses yield inaccurate results. This also means that when a site is well collected and documented, it will skew the analytical results: At Jarmo, northern Iraq, more than 2000 tokens have been found, nearly half (48.7%) of the spheres in SchmandtBesserat’s original dataset.

x

The contexts where small objects made of clay are found may suggest purposes other than counting: For example, being found at a burial site might imply an object was some kind of funerary offerings.25 Such contexts make it possible, if not likely, a token-like object had a use other than counting. Sometimes the contexts aren’t well documented, or the proveniences of artifacts are unknown or suspicious. In these cases, there is no contextual information to help establish an object’s purpose.

x

Particular types of contexts may suggest token function: Finds from Tell Sabi Sbyad, ‘a late Neolithic village in north-central Syria’ that appears to have been used for storing and processing grain and manufacturing cloth, suggests accounting related to those commodities.26 However, contexts are often ambiguous: Tokens found in the rubble fill of a wall that may have once been located near a temple are not convincingly related to the administrative purposes likely to have been performed there. To date, no convincingly unambiguous contexts have been published, though many have been proposed, and there is a general sense that unless contexts unambiguously indicate administration, the identification of the token-like objects found there as numerical counters will remain problematic.

x

Objects identified as tokens have been traded illegally, and for that matter, so have bullae and tablets. Illegal trade raises a host of thorny legal and financial issues regarding publication and use of any data related to an artifact, regardless of how unambiguous its use, context, or provenience might happen to be.

Schmandt-Besserat, Before Writing. Englund, ‘Review: Denise Schmandt-Besserat, How Writing Came About’. 26 Robson, Mathematics in Ancient Iraq, p. 34. 24 25

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The division of tokens into multiple systems, as well as the regional and temporal variation in token shapes and sizes,27 mean there could not have been a single system of numerical meaning for even the plain tokens.28

x

The best source of numerical correspondences for token meaning, impressed bullae and clay envelopes associated with token assemblages, are relatively scarce. When these containers are found unsealed or broken open, they are often not clearly associable with any tokens found nearby.29 Eighty of the known artifacts remain intact, nearly two-thirds of those found to date.30 There is a considerable and frankly commendable reluctance to open them, given the process of accessing the contents destroys the container. Various imaging techniques, while not invasive, have made little progress in identifying contents for comparison to surface marks.31 And as Robert Englund has noted in several of his publications with a palpable dismay, bygone custodians had a habit of shaking sealed artifacts to show there was something inside them. This undoubtedly damaged the fragile contents of unfired clay, to the point where it would be surprising indeed if any, were they ever opened, remained sufficiently intact to compare them to external markings.

Given these many challenges, the prospects for identifying the numerical meaning of tokens before the 4th millennium BCE are rather grim: As Damerow put it, ‘all attempts to identify the measuring and counting units represented by the pre-literate tokens and impressions have failed so far’.32

NEWLY CATALOGUED TOKEN FINDS AND THEIR ANALYSIS I hoped I might resolve at least some of the ambiguity related to the numerical meaning of plain tokens before the 4th millennium BCE by examining finds documented in the decades since Schmandt-Besserat published her comprehensive catalogue. Both she and the University of Texas at Austin Press granted me permission to convert her original dataset to an electronic format for analysis. To this, I added over 2300 new entries from various sources, many of them pointed out to me by Schmandt-Besserat. Sadly, this number represented less than half of the token data potentially available. Several particularly large token finds have been reported but remain unpublished and inaccesFriberg, ‘Preliterate Counting and Accounting in the Middle East’, pp. 483–484. Englund, ‘Review: Denise Schmandt-Besserat, How Writing Came About’; Friberg, ‘Preliterate Counting and Accounting in the Middle East’. 29 Damerow and Meinzer, ‘Computertomografische Untersuchung Ungeöffneter Archaischer Tonkugeln aus Uruk, W 20987,9, W 20987,11 und W 20987,12’. 30 Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’; Friberg, ‘Preliterate Counting and Accounting in the Middle East’. 31 Damerow and Meinzer, ‘Computertomografische Untersuchung Ungeöffneter Archaischer Tonkugeln aus Uruk, W 20987,9, W 20987,11 und W 20987,12’; Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’. 32 Damerow, ‘Prehistory and Cognitive Development’, p. 24. 27 28

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sible to scholars, including 1100 tokens from Tell Tayinat, Turkey33 and 1600 tokens from Tell Sabi Abyad, Syria.34 In a few cases, token finds were in the process of being published, but their authors did not respond to my requests to include their data in my analysis. At the very opposite end of the spectrum, John MacGinnis and Timothy Matney of the Ziyaret Tepe Archaeological Project were unreservedly willing to share both their published and unpublished data with me, giving me complete access to their project’s database and providing me with additional details on their finds whenever I requested them. The 2300+ new entries were by no means an exhaustive inventory of all tokens found and published since 1992. They did, however, comprise a representational sample sufficient to demonstrate that tokens have generally continued to be treated much as Schmandt-Besserat described the situation in the early 1990s: Many known finds remain unpublished. Scholars with unpublished tokens often do not share their data. Authors whose published data lack the details necessary to classify their tokens either do not have more detailed descriptions or prefer not to provide them. Contexts are often under-described or over-interpreted. There is still no standard set of criteria for identifying tokens, nor an established typology for classifying them. Finds might be given the generic designation token, rather than more descriptive labels like cone and sphere. Data that might assist classification—images or drawings, measurements such as dimensions and weight, dating estimations—are not routinely provided. Intact bullae and envelopes are still rare; sealed artifacts remain intact, and there has been little progress in discerning their contents with imaging techniques. Finally, there is no central repository for making token data available to scholars, comparable to the way platforms like the Cuneiform Digital Library make cuneiform texts available. In short, while tokens were unlikely to have been a unified, consistent method of representing numbers,35 their variability and the attendant uncertainty about their meaning have been compounded by the way they are typically treated, further reducing an already limited ability to interpret their numerical use before the 4th millennium BCE. It is nonetheless useful to describe the newly catalogued tokens in terms of type, geographic distribution, and temporal span; contrast these characteristics with those of Schmandt-Besserat’s original dataset; and analyze the total data for what they might reveal about the use of plain tokens as numerical counters. As Schmandt-Besserat’s interpretation of complex tokens remains highly contested, they were not a focus in the analysis that follows. Overall, the new tokens conformed fairly well to the findings Schmandt-Besserat published in 1992: The plain types used for numbers—spheres, cones, disks, cylinders, and tetrahedra—remained the most common shapes (Fig. 9.4), consistent with the widely accepted view these were the types used for numeration. Mentioned in MacGinnis et al., ‘Artefacts of Cognition’. Akkermans and Verhoeven, ‘An Image of Complexity: The Burnt Village at Late Neolithic Sabi Abyad, Syria’; Akkermans et al., ‘Investigating the Early Pottery Neolithic of Northern Syria: New Evidence from Tell Sabi Abyad’. 35 Englund, ‘Review: Denise Schmandt-Besserat, How Writing Came About’; ‘Texts from the Late Uruk Period’. 33 34

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Fig. 9.4. Types of tokens. Comparison of tokens by the 16 types established by SchmandtBesserat in her original dataset (SB92; n = 8477), the newly catalogued tokens (New; n = 2352), and the combined data (Total; n = 10,829). Types are listed in order of token frequency from high to low. In both the original and new datasets, the five types of plain tokens thought to have been used for counting were the most common. Image created by the author.

To collect her data, Schmandt-Besserat examined thousands of artifacts in museum collections. In contrast, I classified new token finds through their published images, drawings, and descriptions. This required combining or converting them to SchmandtBesserat’s typology, as follows: x

Some authors publishing token finds had classified their tokens through their own, typically unspecified, criteria. Several of these individualized typologies shared a high degree of commonality with Schmandt-Besserat’s typology, while others diverged from it, occasionally to a large degree. In all cases, I converted authors’ classifications to Schmandt-Besserat’s typology for analysis.

x

Tokens not initially classified by the publishing authors were classified by applying Schmandt-Besserat’s typology to the available pictures, drawings, and descriptions of the artifacts. In several cases, I coordinated these classifications with Schmandt-Besserat, who verified their accuracy.

x

It was generally not possible to make subtype determinations, given that authors didn’t classify subtype or provide the images and other details needed to make such classifications.

As published, Schmandt-Besserat’s original dataset contained 8162 tokens. Once I had converted it to electronic format, I found several minor errors using software functionality not available in the early 1990s. On the whole, the generally error-free condition of her dataset testifies to the care she took in compiling it. I imposed several minor corrections pointed out in various reviews of her book, and added data she hadn’t published in 1992 for copyright reasons. These changes increased the total to 8477

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tokens distributed between 109 sites in five modern countries (Table 9.2): Iran (40 sites), Iraq (41), Palestine (8), Syria (15), and Turkey (5) and spanning the 9th to the 1st millennium BCE (Fig. 9.5). In comparison, the newly catalogued tokens came from 26 sites, many of which were places where tokens had previously been found: Iran (8 sites), Iraq (7), Palestine (1), Syria (4), Turkey (5), and Jordan (1). They were temporally distributed between the 10th millennium BCE and the 1st millennium of the Common Era. Site

Iran SB92 56 2 6 34 3 65 3 24 575 1 13 1 70 81 23 7 2 17 19 61 8

Anau Bampur Belt Cave Chagha Sefid Chogha Bonut Chogha Mish Dalma Tepe Deh Luran Ganj-Dareh Geoy Tepe Hajji Firuz Istakhr Jaffarabad Jeitun KS34 KS54 KS76 Malyan Moussian Seh Gabi Sharafabad Sheikh-e Abad Sialk 8 Sorkh-i-Dom 1 Susa 705 Tal-i-Iblis 39 Tall-i-Bakun 56 Tepe Abdul Hosein 44 Tepe Asiab 193 Tepe Bouhallan 4 Tepe Farukhabad 5 Tepe Gaz Tavila 93 Tepe Giyan 15 Tepe Guran 34 Tepe Hissar 92 Tepe Muradabad 1 Tepe Sarab 340 Tepe Siahbid 3 Tepe Yahya 142 Tulai 7 Zagheh 29 Total (41 sites) 2882 Jordan Site SB92 Es-Sifya Total (1 site) 0

New

238 1143

Total 56 2 6 81 41 135 3 24 1219 1 13 1 70 81 23 7 2 17 19 61 8 11 8 1 705 39 109 86 193 4 5 93 15 34 92 1 340 3 142 7 267 4025

New 78 78

Total 78 78

47 38 70 644

11

53 42

Site Abu Salabikh Arpachiyah Assur Billa Choga Mami Eridu Fara Gird Ali Agha Gird Banahilk Hassuna Ischali Jarmo Jemdet Nasr Khafaje Kish Larsa Maghzaliyah Matarrah M’lefaat Nineveh Nippur Nuzi Qal’at’Ana Ras al ’Amiya Sippar Tell Abada Tell Agrab Tell Asmar Tell es-Sawwan Tell Hyelri Tell Madhhur Tell Oueili Tell Raschid Tell Songor Tell Yalkhi Tello Telul eth Thalathat Tepe Gawra Ubaid Umm Dabaghiyah Umm Hafriyat Uqair Ur Uruk Yarim Tepe Total (45 sites)

Iraq SB92 93 18 18 12 15 3 1 3 3 2022 27 10 60 3 8 1 2 12 26 26 30 1 50 1 7 77 5 8 4 1 5 92 11 485 1 11 26 5 107 809 59 4158

New 74 16

61

1 180

2 4

338

Total 74 93 16 18 18 12 15 3 1 3 3 2022 88 10 60 3 8 1 2 12 26 26 1 30 1 230 1 7 77 5 2 12 4 1 5 92 11 485 1 11 26 5 107 809 59 4496

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Palestine Site SB92 ’Ain Ghazal 12 Beidha 20 Beisamoun 4 Jericho 3 Ktar Tell Kazarei 1 Megiddo 9 Munhatta 1 Tell Aphek 1 Total (8 sites) 51 Turkey Site SB92 Beldibi 11 Cafer Höyük Can Hasan 41 Çatalhöyük Cayönü Tepesi 108 Gritille 33 Hacınebi Suberde 66 Tell Kurdu Ziyaret Tepe Total (10 sites) 259

New 137

137 New 2 11 12 4 490 519

Total 149 20 4 3 1 9 1 1 188 Total 11 2 41 11 108 33 12 66 4 490 778

Site Abu Hureyra Amuq Chagar Bazar Cheikh Hassan Ghoraife Habuba Kabira Hadidi Jebel Aruda Mureybet Ras Shamra Tell Aswad Tell ’Atij Tell Brak Tell Halaf Tell Kannas Tell Ramad Total (16 sites)

Syria SB92 5 17 9 2 11 141 147 11 2 16 320 3 5 58 380 1127

New 17 13

21 86

137

Table 9.2. Tokens by country and site. Schmandt-Besserat’s original dataset (SB92), as corrected, contained 8477 tokens. Over 2300 tokens (New; n = 2352) were added, bringing the full dataset (Total) to 10,829. A further analysis by token shape is provided in Table A.3. Data compiled from multiple sources.

Fig. 9.5. Temporal distribution of tokens. The time period spans the 10th millennium BCE to the 1st millennium CE. Tokens with unestablished or unpublished dating were categorized as Not Specified (NS). The data presented include Schmandt-Besserat’s original dataset (SB92; n = 8477), the newly catalogued tokens (New; n = 2352), and the combined data (Total; n = 10,829). Image created by the author.

Total 22 30 9 2 11 141 147 11 2 16 320 21 89 5 58 380 1264

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When the original and new data were compared, the earliest sites with tokens (Fig. 9.6) remained places whose ‘geographic distribution … along the Fertile Crescent … suggests that the area where [token-based] recording was practiced first coincides with the region in Southwest Asia where plant and animal domestication first appeared’.36 The number of tokens declined after the 4th millennium BCE, consistent with SchmandtBesserat’s conclusions about the availability of writing decreasing the use of tokens. However, finds at Ziyaret Tepe, Turkey from the 1st millennium CE37 suggested the decline in token use was neither as rapid nor as complete as Schmandt-Besserat believed. There are several possible explanations for this. One is that Ziyaret Tepe might be considered peripheral to the region where writing was invented and from which it spread, so it may have been later in adopting the new technology as a function of time and distance. Another is that while impressions and notations are good at representing information, they are fixed forms not particularly well suited for its manipulation, necessitating the continued use of tokens for calculation. In fact, a token-like means of calculation—the abacus—would remain in use as the preferred method of calculating for millennia after tokens disappeared from the archaeological record. Algorithms with written notations emerged subsequent to the invention of writing, ultimately displacing the abacus; though this transition too took millennia.38 Neither has the abacus been completely displaced: It was used in Europe as late as the Middle Ages39 and remains in use in various parts of the world.40 The abacus competed with electronic calculation as recently as 1946,41 and it continues to be investigated for what it reveals about the involvement of finger-movement planning in calculating.42

Schmandt-Besserat, ‘The Emergence of Recording’, p. 872. MacGinnis et al., ‘Artefacts of Cognition’. 38 Reynolds, ‘The Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators’; Stone, ‘Abacists versus Algorists’. 39 Boyer and Merzbach, A History of Mathematics. 40 Donlan and Wu, ‘Procedural Complexity Underlies the Efficiency Advantage in Abacus-Based Arithmetic Development’. 41 Kojima, The Japanese Abacus: Its Use and Theory. 42 Brooks et al., ‘Abacus: Gesture in the Mind, Not the Hands’; Frank and Barner, ‘Representing Exact Number Visually Using Mental Abacus’. 36 37

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Fig. 9.6. Geographic distribution of early tokens. Map showing the geographic distribution of sites from the 10th to the 8th millennia BCE where tokens have been found. Adapted from an image in the public domain.

I also analyzed the five plain token types used in accounting by their amounts and chronology (Fig. 9.7) with the idea this might provide insight into how long a period of numerical use they might have had before the 4th millennium BCE. The analysis showed a roughly similar pattern of increase and decrease for four of the five shapes, with general consistency in their relative proportions: Spheres were the most common, tetrahedra the least. The increase in token finds dated to between the 9th to the 6th millennia BCE was consistent with estimates that the use of numerical tokens emerged at some point during the period. There was a significant decrease from the 4th millennium BCE onward, suggesting that writing replaced tokens for numerical representation at a fairly rapid pace. The small increase in finds dated to the 1st millennium BCE and 1st millennium CE was associated with peripheral locations, perhaps reflecting the uneven pace at which technology would have spread in an era when distance was a considerable barrier to the exchange of ideas. The decrease in frequency around the 5th millennium BCE in four of the five types was difficult to explain, regardless of the social purpose the objects may have actually served. It suggests a common cause, perhaps a population decrease related to warfare or the loss of trade in specific commodities.

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Fig. 9.7. Temporal distribution of plain types used as numerical counters. Five token types— spheres (n = 2233), cones (n = 1533), disks (n = 1159), cylinders (n = 766), and tetrahedra (n = 327); total (n = 6018)—were associated with use as numerical counters, especially from the mid-to-late 4th millennium BCE and onward, as highlighted. Except for cones, the plain types show a roughly similar pattern of increase and decrease. The 8thmillennium BCE cones from Ganj-Dareh (n = 828) and 6th-millennium BCE spheres from Jarmo (n = 1399) were omitted, as the unusually large quantities from these sites skewed the results. Image created by the author.

These findings do not entail the plain tokens in the sample were all numerical counters. This seems unlikely in any case, given the different issues perplexing the identification of objects as tokens and the span of time involved. Certainly, cones differed in their temporal distribution from the other types, which is interesting, given that cones were integrated into multiple numerical systems in the 4th millennium BCE, implying their prevalence should have been both substantial and consistent. The fact it was not suggests cone-shaped objects may have had non-numerical purposes at a rate higher than that of the other shapes, and other explanations—selection bias, misclassification— seem possible. From the available evidence, I concluded there was still no way to tell when token-based accounting really emerged or how consistent it may have been over time. However, these issues do not need to be resolved to recognize tokens as an important transitional technology, one bridging one-dimensional token precursors like tallies and the tokens’ two-dimensional successors, the numerical notations. As such, tokens would likely have shared characteristics with one or the other, perhaps more tally-like and used in the one-to-one correspondence envisioned by Schmandt-Besserat in earlier periods, more notation-like and functioning like an abacus in later ones. With an abacus-like functionality, calculating with tokens would have been a matter of physically moving counters, and the mental knowledge required in calculating would have consisted of understanding how the counters were to be moved and exchanged. Yet tokens also involved an expanded set of relations between numbers and arithmetical operations, relative to precursor technologies. Where the notches on a tally had relations like more than and one more, tokens were related by six and ten, and they could also be related by two, three, or four. Thus, tokens represent significant steps toward the development of a numeracy involving, and depending upon, mental knowledge.

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THE COMPLEXITY OF TOKENS AND TOKEN-BASED ACCOUNTING For at least a short span of time before invention of writing, plain tokens are reasonably assumed to have been used like the early numerical impressions were, given the correspondences between the two attested by token-shaped impressions in clay: Any particular token would have taken its numerical value from its shape, size, and context of use. Collections of tokens would have been ordered by the magnitude of their value, from least to greatest. The type of commodity being counted would have been indicated through specific combinations of token shapes, sizes, order, and numerical relations, though other methods would have been used as well: modifying plain tokens, including complex tokens, and marking bullae externally. Tokens not otherwise marked or contextually specified might also have been exchanged between trading partners whose identity specified commodity, though this wouldn’t be visible to us from the material remains. How commodity and quantity were conjoined is shown in the top block of Fig. 9.1, which shows tokens with the same shapes and sizes used for counting grain and animal products. Specific combinations of shapes, sizes, and relations indicated numerical value and commodity. As represented in the figure, the two Mesopotamian systems shared three of four shapes and sizes, as well as their ordering, so these qualities would not differentiate the two systems. However, the disk meaning 1500 liters, because it was unique to the system for counting grain, would identify the commodity being enumerated as grain. Tokens could be polyvalent: In the system for counting grain, a large cone meant a container of 4500 liters equivalent to three disks meaning 1500 liters each, while in the system for counting animals and animal products, a large cone had the value 60 because it was equivalent to six small spheres meaning 10 each. Within the system for counting grain, a small sphere could mean 5 or 150 liters, depending on whether it preceded or followed a cylinder. Such ordering is easily fixed when tokens are impressed into clay, but we don’t know what conventions might have been used with loose tokens. In other number systems, this has been solved through color encoding or physical separation: In the Incan quipu, the way knots were tied and where they were placed along a string signified quantity, while color and position in the series of strings signified commodity.43 Today their descendants, Peruvian herdsmen, ‘count their beasts by means of beans in a small bag which they always carry with them; the colours of the seeds correspond to the various kinds of animals, cows, calves, and so forth’.44 Such strategies involve materials that tend to be highly perishable, leaving few archaeological signs for us to find. As attested by later notations, the rules used in token-based accounting ‘required that multiples of a given unit … be replaced by the next “higher” units’.45 Simply, groups of tokens would have been combined or separated and then sorted; once this had been done, units with higher and lower values would have been exchanged to simplify results to the lowest, most intelligible terms. That is, the quantity of twenty tokens Urton, ‘Recording Measure(ment)s in the Inka Khipu’. Baudin, Daily Life in Peru: Under the Last Incas, p. 39. 45 Friberg, ‘Preliterate Counting and Accounting in the Middle East’, pp. 483–484. 43 44

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worth a unit each is more difficult to understand than that of two tokens worth ten units each, even though the amounts are equivalent. This is because the subitization limit makes it difficult for us to appreciate quantities higher than about three or four, and because the ability to appreciate the magnitude of quantity differences between groups lacks the precision desirable for accounting (Chapter 4). This visual indistinguishability is true of fingers and tallies as well (Chapter 8), though with these technologies, the inconvenience is perhaps lesser because the quantities involved are typically smaller. As the quantities being represented and manipulated increase with the capacity of the number system, however, there is an increased pressure to solve the problem of visual distinguishability. Across number systems and as was true of the tokens, this is typically accomplished by grouping or bundling the material exemplars of quantity. Bundling, the second dimension of exponential representation, often involves quantities like five and ten evoking the fingers, but it can also involve quantities based on metrological units, like four quarts equal a gallon in modern liquid measures. Tokens and the written notations following them appear to have incorporated bundling amounts from both sources. Tokens solved another problem associated with tallies and many other onedimensional devices: their lack of manipulability. Notches on a tally are difficult, if not impossible, to move or remove, except in the limited sense of planning where they are to be placed during the process of making them. Tokens, in comparison, are not fixed, so they can be combined, recombined, and removed. If the oldest economic tablets, and presumably, the tokens preceding them as well, did not ‘contain a single example of a genuine multiplication’,46 combining and removing tokens and then simplifying them by exchanging multiple tokens of smaller values for single tokens of higher value, or vice versa, certainly comprised a functional addition and subtraction. Similarly, moving half to one side would have comprised a functional division, doubling or replacement a functional multiplication. Implicit to multiplication—to all of the arithmetical operations, really—is the idea numbers are numerically related. It’s difficult to multiply 5 by 6 if you don’t know that 5, 6, and 30 are related. Certainly, these concepts are implicit to the highly elaborated numbers of the Western numerical tradition, as I explained in Chapter 3. But I contend the concept of number does not come fully loaded with these relations when it first emerges. Rather, like numbers themselves, the relations between numbers must be patiently worked out. This explication and elaboration take time—generations of it—and this in turn requires a social context where using numbers is prioritized to the extent multiple generations remain involved in using them. It also requires incorporating different forms of materiality, ones with properties like manipulability that provide opportunities and occasion the insights allowing relational explication and elaboration. The Mesopotamian tokens provided just such an opportunity, and the exchange operations show their association with a set of relations expanded well beyond those of onedimensional technologies. That is, the idea six or ten of these mean one of those, implicit to 46 Damerow, Abstraction and Representation: Essays on the Cultural Evolution of Thinking, p. 236; emphasis in original.

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THE MATERIAL ORIGIN OF NUMBERS

the tokens’ bundling conventions and exchange operations, is difficult to realize with a fixed representational method like the tally. Exchange operations imply some method of sorting and organizing groups of tokens by value, as this would facilitate exchange operations. As their impressions in clay attest, this organization took the form of arranging tokens in order of their value, from low to high. Ordering tokens by value would also have been consistent the tendency to order signs from high to low in ‘over 100 different numerical notation systems spanning over 5000 years and every inhabited continent’, with only minor exceptions occurring in ‘certain alphabetic systems’.47 Conversely, the hypothetical lack of any method of separating tokens by type—or more importantly, arranging them in order of magnitude—would have made exchange operations difficult to perform, if not more prone to errors. Similarities between calculations from the Old Babylonian period (1900–1600 BCE) and developments in much later periods suggests tokens were used for calculating and that this involved a counting board to organize them.48 Counting boards are not attested archaeologically, presumably because they were made of wood and didn’t preserve, the same issue we saw with early tallies in the region. Neither are they depicted in reliefs, nor are they described textually. Counting boards are, however, suggested by artifacts identified as game boards,49 as these have potential utility in calculating,50 as well as the visual appearance of later cuneiform signs: The cuneiform sign ŠID, ‘an ideogram with the meaning “account”’,51 count(ing), and number,52 may have originally depicted a device like an abacus (Fig. 9.8).53 Whether this vague resemblance means the abacus truly originated in Mesopotamia, as mathematical historian Georges Ifrah asserted,54 such devices and strategies would have facilitated both exchange operations and the eventual development of place value as well. However, the idea that an abacus or counting board was used has not been universally accepted.55 As mentioned earlier, exchange operations also imply the availability of, and some faculty with, an ordinal counting sequence extending well into the non-subitizable range, as this would be essential to performing exchanges. If there were no ordinal counting sequence or faculty using it, exchange operations would have been difficult— if not impossible—at the quantities found in the period before writing, which reached several hundreds. An alternative to using an ordinal counting sequence might be estiChrisomalis, Numerical Notation: A Comparative History, p. 364. Høyrup, ‘A Note on Old Babylonian Computational Techniques’; also see Friberg, ‘Methods and Traditions of Babylonian Mathematics: Plimpton 322, Pythagorean Triples, and the Babylonian Triangle Parameter Equations’. 49 Rollefson, ‘A Neolithic Game Board from ʿAin Ghazal, Jordan’; Simpson, ‘Homo Ludens: The Earliest Board Games in the Near East’. 50 Manansala, ‘Sungka Mathematics of the Philippines’. 51 Nissen, Damerow, and Englund, Archaic Bookkeeping, p. 134. 52 University of Pennsylvania, The Pennsylvania Sumerian Dictionary. 53 Smith, The History of Mathematics, Vol. 1, p. 40. 54 Ifrah, The Universal History of Computing: From the Abacus to the Quantum Computer. 55 Oppenheim, ‘On an Operational Device in Mesopotamian Bureaucracy’. 47 48

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mating the respective sizes of piles of tokens, though this would lack the computational accuracy generally characterizing the earliest numerical impressions. Another alternative might be exchanging tokens on a one-to-one basis, though this would be cumbersome and laborious. Were they used in actuality, the drawbacks and deficiencies of both would have exerted pressure to achieve greater efficiency and effectiveness, particularly as requirements for using numbers increased along with increases in population, trade, construction, labor pools, and urban size. An inefficient, ineffective alternative is unlikely to have persisted as such, especially for very long.

Fig. 9.8. The cuneiform sign ŠID. Used between 3000 and 2000 BCE,56 the sign’s meaning— account, count(ing), number—and vague resemblance to a counting board or abacus has been a longstanding basis for believing such devices may have used in the Ancient Near East. The word abacus is thought to have originated as the Semitic word abq, meaning sand or dust, suggesting an Ancient Near Eastern origin for the device as temporary lines drawn to separate numerical counters ordered by their value. Historian Alfred Nagl observed that as moving free-standing counters would quickly blur or erase such temporary lines, the idea of an ancient dust-board had to be dropped, but he also noted lines were not essential to ordering and exchanging counters during calculation.57 In any case, lines would be more critical for operations with indistinguishable counters used for different values, less so for tokens. The image is a Unicode character.

If their manipulability enabled exchange operations, tokens came with significant drawbacks as well: First, their conventions of size and shape didn’t encode quantity alone, but quantity conjoined with commodity. Being conjoined not only made the conventions more complex, the material form couldn’t represent and manipulate numbers by themselves. Second, tokens were loose, and loose materials, whether used for calculating or recordkeeping, are relatively easy to disturb and alter. This meant tokens needed to be contained, especially as their use proliferated, and this motivated the development of bullae, surface impressions on bullae, and tablets with impressions. The notations that would develop from this technological sequence not only made tokens redundant as a method of recordkeeping, they would allow the conjoined representation of commodity and quantity to be separated. This technological change led to writ56 57

University of Pennsylvania, The Pennsylvania Sumerian Dictionary. Nagl, ‘Abacus (Supplement to Original 1893 Article)’.

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ing and literacy on the one hand, and an elaborate mathematics on the other, 58 important developments described in the next chapter.

58 Englund, ‘Texts from the Late Uruk Period’; Malafouris, ‘Grasping the Concept of Number: How Did the Sapient Mind Move beyond Approximation?’; Schmandt-Besserat, Before Writing.

CHAPTER 10. NUMERICAL NOTATIONS AND WRITING How writing changed ancient numbers starts with pointing out the two things involved when someone reads the words on this page. First and rather obviously, the reader must be literate, able to read and write with some proficiency. Literacy involves learning to recognize the characters used to form words and sentences and becoming able to reproduce them with pencil and paper. Literacy has a psychological component: Recognizing characters and associating them with the meanings and sounds of language imply a neurological reorganization, one understood fairly well by neuroscience at this point. There is a behavioral component in things like moving the eyes across the page, controlling the fine motions needed to produce legible characters, and coordinating the movements of hand and eyes. There are social and pedagogical conditions as well, which determine access to learning, opportunities to practice enough to gain proficiency, protocols for qualifying and selecting instructors, and theories and methods of instruction. Without the requisite conditions and changes, written words are just slightly more intelligible than chicken scratches—recognizable for their regularity, but just as meaningless. The second thing involved when someone reads these words is less conspicuous: It’s what the material form of writing itself does during the activities described in the previous paragraph, and how it becomes able to do the things it does. We tend not to think of writing as an active participant in cognitive states like reading and literacy. Rather, we conceive writing in passive terms, as something that doesn’t change except as we produce it when we write. We focus on its symbolic nature—forgetting it’s also a material form that has changed greatly over the several thousand years we’ve been engaged with it, and one that has become increasingly adept at expressing language with fidelity and creating behavioral and psychological changes in those who interact with it. And we tend to think of it as something invented once upon a time, a tool created by our human ingenuity—and not as a material form whose function and effectiveness emerged from, and were refined and intensified through, generations of interactions with it, a process lacking both teleological purpose and goal. Some of this material inconspicuity can be attributed to the fact that writing doesn’t change much during a human lifetime. I don’t mean that there are no differences between a chalkboard, the printed page, and a computer screen or that the differences are unimportant. It’s just that the forms of the characters themselves don’t change much across these different media, except in the sense of an individual scrawl, the smorgasbord of fonts, and the specific motor movements involved in handwriting 179

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and typing. Character forms themselves have changed much more slowly—over centuries, if not millennia—and often in unintentional, non-deliberate ways. This makes it difficult for psychology and neuroscience to address the material form that is writing: Their theories and methods are geared toward stable stimuli, controlled change, and shorter spans of time. Another portion of inconspicuity can be ascribed to our thinking of writing in terms of what it means—its content, translation, or symbolic value; the significance and implications of its context or authorship—or the deeply philosophical question of why it means. These things I leave in the capable hands of paleographers and semioticians, for as a cognitive archaeologist, I am interested in how writing means as a material form and the process whereby it becomes more expressive of meaning. This chapter explores how early Mesopotamian writing, whose expressiveness was so limited scholars still debate which language its inventors would have spoken,1 became the cuneiform script, a writing system capable of supporting a literacy analogous to what we mean by the term. In this analysis, writing is treated as a material form tractable to cognitive-archaeological methods, one that attests to behaviors like writing and provides insight into psychological processes like language, character recognition, and automaticity. Also examined are how numbers changed through being written and how writing numbers in Mesopotamia enabled their elaboration as one of the ancient world’s great mathematical traditions. And it looks at how written numbers differ from written non-numerical language—instantiating their meaning, for example, rather than signifying through resemblance or convention—giving them a perhaps surprising contiguity with non-written precursors like tokens and tallies.

FROM TOKENS TO IMPRESSIONS The earliest numerical impressions on bullae are assigned to the Pre-Uruk V period (8500–3500 BCE). These were followed by tablets with numerical impressions assigned to the Uruk V period (3500–3350 BCE). The Uruk periods are paleographic designations, not chronologies established by archaeological dating methods, and artifacts are assigned to them through critical evaluations of the type of writing they bear. As I mentioned in the last chapter, these earliest impressions looked a lot like their predecessors, the tokens. In fact, many appear to have been made with the tokens themselves.2 It’s the resultant correspondences of size, shapes, and quantities between the two that tell us tokens had numerical meaning, at least in period where the different forms—tokens, bullae, and tablets—overlap. In 2015, the online database of the Cuneiform Digital Library listed 193 tablets from the Uruk V period with at least one legible numerical impression on them (Table A.2). Most of these tablets were relatively simple, consisting of a few numerical values disposed across a limited number of cells or columns. About 40% contained legible impressions of more than one numerical value, with N14 and N01 being a frequent 1 Englund, ‘Texts from the Late Uruk Period’; Hyman, ‘Of Glyphs and Glottography’; Veldhuis, History of the Cuneiform Lexical Tradition. 2 Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’, p. 23.

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combination. This percentage is likely an underestimation, given that many signs are illegible and the artifacts are often incomplete, so signs are undoubtedly missing. The signs that are present and legible, however, are sufficient to show the numerical impressions were ordered by increasing numerical value (Fig. 10.1).

Fig. 10.1. Magnitude ordering of numerical signs. (Top) Administrative tablet (JA 74-024) from Jebel Aruda assigned to the Uruk V period, most recently held in the collection of the National Museum of Syria at Raqqa. Its impressions—two N34, one N45, seven N14, and two N01—were ordered by increasing magnitude. (Bottom) Corresponding signs in the ŠE System Š for grain. The artifact likely represented grain in ŠE System Š, since in all the other counting systems in which N14 appears, its unbundled maximum is five, except for GAN2 System G, which does not include N34. The image of the Jebel Aruda artifact is from Van Driel, ‘Tablets from Jebel Aruda’, p. 20, Fig. 4a, and used with permission from Brill. The notations in the ŠE System Š were redrawn from Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East, p. 29, Fig. 28.

Where there were legible signs of different value on the same artifact, the repetition quantities conformed well to the rules for bundling and exchange determined by Hans Nissen and his colleagues.3 There were a few exceptions: The seven artifacts listed in 3 Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East.

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Table 10.1 had signs whose repetition exceeded their unbundled maximum, the highest number of signs possible without being exchanged with one of the next higher value; it is equal to the exchange value minus one. For example, the unbundled maximum for N14 is nine, since ten N14 equal one N50 in the GAN2 System G for measuring area or one N45 in the ŠE System Š for grain. Violation of the bundling rules is often highlighted as evidence their users didn’t completely understand what they were doing, numberwise. Of note, all eight instances exceeding the unbundled maximum in the sample involved N14. This possibly reflects the fact that many of the tablets assigned to the Uruk V period do not contain legible signs for higher quantities; if they did, we would be better able to assess their unbundled maximums, if any, and determine if these were exceeded at comparable rates. It’s also possible the exceeded bundling reflected specific purposes, like area calculations. In one of the eight cases, the N14 signs lined the edges of the artifact, forming a quadrilateral (Fig. 10.2, left). There are several other cases where the same surface disposition was used, though the others fall within the unbundled maximum for N14, and all have been interpreted as area calculations. Source Designator Unbundled Tell Jokha

MS 4648

Susa

(P008553)

Tepe Sialk

Sialk 1631

23 N14

Jebel Aruda

JA 74-020 JA 75-104 JA 75-105

10 N14 22 N14 10 N14

Mari

T 084

11 N14

12 N14 14 N14 (two instances)

Disposition Two rows of 6 N14 each Two rows of 14 N14 each Two rows of 10 and one row of 3 Two rows of 5 Two rows of 11 Two rows of 5 Three rows of 4, 6, and 1 N14

Adjacent Signs Higher Lower

Figure







4 N14 (top)

3 N14 (bottom)

10.3, left







1 N34 3 N34 —

3 N01 5 N01 —

— 10.3, right —

[broken]

3 N01



Table 10.1. Uruk V numerical tablets with exceeded bundling. All instances exceeding the unbundled maximum in the sample involved N14, whose unbundled maximum is nine in GAN2 System G (area) and ŠE System Š (grain). Three of the instances (JA 74-020, JA 75-105, and T 084) were at or slightly over the bundling limit, while two (Sialk 1631 and JA 75-104) exceeded the limit by more than double. The CDLI designator is given in parentheses when an artifact is unpublished or has not been assigned a museum, accession, or excavation number. The data were drawn from the Cuneiform Digital Library on 14 August 2015.

As for some of the other artifacts with too many N14 on them, alternative explanations seem possible. If the N14 didn’t represent an area, their placement in a linear column might still have had a spatial meaning we fail to understand. They might have represented the exact containers in the one-to-one fashion Schmandt-Besserat proposed. Or they might simply have been added later than the surrounding ones. Consider the artifact shown in Fig. 10.2, right. If the quantity had been bundled as expected, it would have appeared as four N34, one N14, and five N01, not three N34, eleven N14, and five N01. Exchanging signs after they were made, however, would have been difficult. Clay is a bit more forgiving than the surface of a tally when it comes to erasing signs, but it’s still a lot less amenable to correction than tokens were. Certainly, clay still malleable enough to add impressions should permit their erasure as well. However, erasure risks

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interfering with adjacent signs, something undesirable when the intent is not to disturb them. Given a choice between muddling the adjacent signs, starting over with a new tablet, or adding excess signs with the understanding the quantity being expressed was nonetheless intelligible, the third option, though inelegant, might have been occasionally preferred. I don’t suggest we assume this is what happened, since there’s really no way to tell, and we don’t want to overestimate ancient numeracy. At the same time, assuming numerical incompetence as the default position risks underestimating it, something we might also want to avoid.

Fig. 10.2. Uruk V artifacts with higher-than-expected N14 repetition. (Left) Administrative tablet (P008553) from Susa, provisionally assigned to the Uruk V period, current location unknown. The artifact contains two rows of 14 N14 each and has been interpreted as an area calculation because the signs are arranged as two sides of a quadrilateral. (Right) Administrative tablet (JA 75-104) from Jebel Aruda, assigned to the Uruk V period and most recently held in the collection of the National Museum of Syria at Raqqa. The artifact contains 22 N14, which might also be read as two rows of 11 N14 each. Image of the Susa artifact is from Scheil, Mémoires de la Mission Archéologique de Perse, Mission en Susiane, Vol. 17, first published in 1923 by Éditions Ernest Leroux and reproduced with the kind permission of Presses universitaires de France. The image of the Jebel Aruda artifact is from Van Driel, ‘Tablets from Jebel Aruda’, p. 21, Fig. 6, and used with permission from Brill.

Whatever their true explanation, what’s interesting about these unbundled representations is the relatively circumscribed range of their variability. That is, there aren’t artifacts with too many N01 signs, something we would be able to see in the available data on legible signs. There aren’t ones with N14 signs sprinkled across their surfaces. Rather, N14 signs are placed at the edges of the face, a disposition suggesting they relate to an area calculation, or they form a column of signs placed centrally on the obverse surface, a disposition whose purpose is unknown but which seems intentional. Given the wide geographic distribution of sites where these artifacts were found (Fig. 10.3), this relative consistency suggests the unbundled signs might indeed have been intentional.

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Fig. 10.3. Geographic distribution of numerical tablets assigned to the Uruk V period. The new technology—writing—emerged in Mesopotamia’s burgeoning urban areas in the mid-tolate 4th millennium BCE. Artifacts with unbundled N14 signs have been found at five of the 13 sites. Adapted from an image in the public domain.

FROM IMPRESSIONS TO COMMODITY LABELS Once clay surfaces appeared, their potential to contain information beyond numerical impressions was quickly explored and exploited. Writers soon realized they could draw small images to label the commodity being enumerated (Fig. 10.4). These images consisted of two types, pictographs and ideographs (Fig. 10.5), respectively, ‘figurative and conventional drawings intended to have a communicative function’.4 Pictographs expressed a range of meanings through resemblance, so a picture of a head could mean things like head, person, or capital. Ideographs expressed meaning through conventions, so a quartered circle meant sheep or ungulate because everyone knew that’s what it meant. We use these representational modes today in things like desktop icons and road signage. In contrast, numerical signs were meaningful because they instantiated quantity, so three N01 signs meant three, as three small cone-shaped tokens had before them, because there were three of them. Instantiation was unambiguous regarding quantity but didn’t convey any information about what was being counted. In tokens, commodity had been expressed through conventions of shapes and sizes, and perhaps specified further with added markings. With tablets, commodity was labeled with pictographs or ideographs. Bundled values were still expressed conventionally, as they had been with tokens: A large cone-shaped impression meant six small spherical impressions, and

4

Glassner, The Invention of Cuneiform: Writing in Sumer, p. 84.

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they in turn meant ten cylindrical impressions, in the same way a large cone-shaped token had been equivalent to six small spheres, a small sphere to ten cylinders. Meaning expressed through resemblance, convention, or instantiation differs in terms of its ambiguity. Instantiation is relatively unambiguous in representing quantity, since three of something simply is three. When the quantity is higher than the subitizing range of about four, its appreciability depends on being able to count. Resemblance, in comparison, is ambiguous in that it doesn’t specify a particular word. Rather, each pictograph has a range of potential meanings, though the range is reasonably limited to meanings related to the object being depicted.

Fig. 10.4. Administrative tablet (W 6066,a) from the city of Uruk. The tablet is assigned to the Uruk IV period (3350–3200 BCE) and was at one time held at the National Museum of Iraq at Baghdad. It shows the separation of numerical and commodity information as different types of signs organized into cells demarked by horizontal and vertical lines. For example, the cell at the top of the left column contains two N14 signs (small circular impressions) and the sign for sheep (a quartered circle). The image is used by permission of Gebr. Mann Verlag.

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Fig. 10.5. Representational modes of signs in early writing (ca. 3200 BCE). (a) Pictographs signified by resemblance. (b) Numerical signs instantiated quantity. (c) Ideographs and (d) numerical bundling signified through convention. Images (a) and (c) adapted from The Pennsylvania Sumerian Dictionary; (b) and (d) redrawn from Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East.

Conventions used in signs for non-numerical language depended on cultural knowledge. When this is absent, as it necessarily is for long-extinct societies, the meaning of such signs is inferred from other matters—the contexts in which the signs are used; the number of times they might be repeated within and across artifacts and texts—often without establishing meaning to the desired level of certainty. Conventions in numerical bundling, which make one sign equal to a number of other signs, are generally unambiguous. This is because values can be determined by analyzing sign contexts and repetitions and because numerical representations have relatively few things needing conventionalization: value, accumulation, and grouping. These things can only be organized in a few ways; linguistic anthropologist Stephen Chrisomalis estimates there may be as few as five basic types among the more than one hundred number systems existing today or known historically.5 This is quite distinct from language, where no two are the same, even those closely related. And in turn, given an unknown script or language, this is why numerical representations are fairly easy to translate and the rest of the signs are not. Thus, even at the point where writing first began in Mesopotamia, there were substantial differences between written numbers and writing for non-numerical language. Besides differences in representational modes and the associated ambiguity, numbers, by this point, had been represented materially for several thousand years—from the beginning of the Neolithic, if we look only at tokens; from the late Upper Paleolithic, if we consider archaeological evidence of possible tallies and linguistic evidence of things like finger-counting. Language, in comparison, had never taken material form before, 5

Chrisomalis, Numerical Notation: A Comparative History.

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except in the limited sense of ownership markings applied as seals, a practice that had emerged in the 6th millennium BCE.6 Numerical elaboration had been a matter of recruiting new material forms, with the properties of those forms, and the habits and knowledge acquired from using them, making numbers linear, ordered, and twodimensional. The elaboration of written language would not be a similar matter; rather, a single form—the set of written characters—would undergo incremental modification and change as it was used over successive generations of time. The amount of ambiguity inherent in the different representational modes would also influence how the different types of signs would change, something I’ll discuss in greater detail toward the end of the chapter. Signs for quantity, especially subitizable quantity, would change little. This was because they unambiguously instantiated their semantic meaning and did not need phonetic specification, the semasiography noted earlier. In contrast, signs that signified language through resemblance or convention changed a lot over the 15 centuries separating the emergence of writing from the development of literacy. Their ambiguity put non-numerical signs under pressure to specify the words intended, and specification involved altering their visual appearance to incorporate determinatives, clues to the kind of word meant, and phonography, techniques for representing pronunciation.

HOW EARLY MESOPOTAMIAN WRITING BECAME THE CUNEIFORM SCRIPT One of the most dramatic changes in the visual appearance of non-numerical signs is their becoming less recognizable as the pictures they began as. Schmandt-Besserat characterized this change as going from concrete to abstract: ‘Hypotheses about the origin of writing generally postulate an evolution from the concrete to the abstract: an initial pictographic stage that in the course of time … becomes increasingly schematic’.7 This characterization carries the Piagetian baggage that abstract is better than concrete and that a meaningful distinction can be drawn between the two. Given that resembling differs from instantiating, this particular change in visual appearance might be more accurately characterized as an increasing independence from the objects depicted, the sense used in the term abstract art. Schmandt-Besserat also speculated ‘the carelessness of scribes’ may have contributed to the increasing abstractness of the character forms.8 Other scholars believe the later cuneiform characters were invented separately from the earlier pictographic and ideographic signs,9 pointing to their distinct iconographic conventions as evidence of this. Conventional differences, however, don’t necessarily preclude the possibility the later signs developed from the earlier forms. The two also aren’t related by an unbroken lineage. Writing largely disappears between the Jemdet Nasr (3200–3000 BCE) and Early Dynastic IIIa (2600–2500 BCE) periods, and the first forms to reemerge after this lacuna are a bit cruder in appearance and less diverse as a set, properties indicative of a Shendge, ‘The Use of Seals and the Invention of Writing’. Schmandt-Besserat, ‘The Earliest Precursor of Writing’, p. 50. 8 Schmandt-Besserat, p. 50. 9 Glassner, The Invention of Cuneiform. 6 7

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loss in continuity. Yet these later forms are not wholly independent of the earlier ones; rather, both sets of characters depict much the same objects in much the same way (Fig. 10.6). None of these explanations strikes me as particularly compelling. Chronologies of change in signs unambiguously relate later forms to earlier ones, and writing on clay requires a certain deliberation in how characters are made if the goal is legibility, the antithesis of carelessness. Loss of depictiveness can also be explained neurologically in a way that yields a system of contrastive graphic elements, a quality that differentiates writing from script: The change in written form, in conjunction with other changes in the writing system, show the emergence of literacy from handwriting behavior.

Fig. 10.6. Chronology of signs. Few examples of writing are known from the period between the Jemdet Nasr and Early Dynastic IIIa periods. However, there is clear continuity between signs before and after the gap. By the Early Dynastic IIIa period, characters were less depictive of the objects they had resembled in the Jemdet Nasr and earlier periods, and as a set, they were much more alike to one another than the earlier pictographs and ideographs had been. By the Old Babylonian (1900–1600 BCE) and Middle Assyrian (1400–1000 BCE) periods, characters had simplified. Redrawn from Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East, p. 111, Fig. 88.

The description of how this occurs is based on the neuroscientific understanding of literacy, pedagogical insights into handwriting, and an archaeological analysis of writing as a material form whose change in form implies change in behaviors and brains. It focuses on how writing and literacy relate to mathematical elaboration, since numbers are my present concern. Readers interested in greater detail on how I see literacy

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emerging from the material engagement of writing might consult my other publications on the topic.10 Literacy entails a specific neurofunctional reorganization in the brain. The region of the temporal lobe involved in recognizing physical objects, the fusiform gyrus, becomes trained to recognize written characters as if they were physical objects.11 Once this occurs, this region is called the Visual Word Form Area. This training effect is thought to be an instance of neuronal recycling, in which a part of the brain with an evolutionary function for responding to natural stimuli has sufficient plasticity to respond to, and become co-opted for, cultural stimuli like written marks.12 Neuronal recycling is assessed by identity in neuroanatomical activation, as well as similarity of functions and constraints in the way it operates with natural and cultural stimuli. The fusiform gyrus also interacts with Exner’s Area, a region of the frontal lobe implicated in planning and executing the motor movements specific to handwriting,13 and Broca’s and Wernicke’s Areas in the frontal and temporal lobes, long known to be the brain’s main centers for producing and comprehending speech.14 This neurofunctional reorganization appears common to all writing systems, whatever their physical form and regardless of whether signs map to words, syllables, or sounds.15 There are, however, distinct patterns of neural activity when characters are visually complex, like they are in Chinese and presumably would be in cuneiform. These suggest differences in working memory16 and phonological processing,17 functions reasonably related to the additional demands imposed by the greater complexity of the characters.

10 Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’; ‘Thinking Materially: Cognition as Extended and Enacted’; ‘Updating the ‘Abstract–Concrete’ Distinction in Ancient Near Eastern Numbers’. 11 Cohen and Dehaene, ‘Specialization within the Ventral Stream: The Case for the Visual Word Form Area’; Dehaene, The Number Sense: How the Mind Creates Mathematics; McCandliss, Cohen, and Dehaene, ‘The Visual Word Form Area: Expertise for Reading in the Fusiform Gyrus’; Vogel, Petersen, and Schlaggar, ‘The VWFA: It’s Not Just for Words Anymore’; also see Price and Devlin, ‘The Myth of the Visual Word Form Area’. 12 Dehaene and Cohen, ‘Cultural Recycling of Cortical Maps’. 13 Roux et al., ‘The Graphemic/Motor Frontal Area Exner’s Area Revisited’. 14 Pegado, Nakamura, and Hannagan, ‘How Does Literacy Break Mirror Invariance in the Visual System?’; Tremblay and Dick, ‘Broca and Wernicke Are Dead, or Moving Past the Classic Model of Language Neurobiology’. 15 Bolger, Perfetti, and Schneider, ‘Cross-Cultural Effect on the Brain Revisited: Universal Structures plus Writing System Variation’; Carreiras, Duñabeitia, and Perea, ‘Reading Words, Numb3r5, and $ymßol$’; Frost, ‘A Universal Approach to Modeling Visual Word Recognition and Reading: Not Only Possible, but Also Inevitable’. 16 Cantlon and Brannon, ‘Adding up the Effects of Cultural Experience on the Brain’. 17 Tan et al., ‘Neuroanatomical Correlates of Phonological Processing of Chinese Characters and Alphabetic Words: A MetaǦanalysis’.

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Words are recognized, as objects are, through the cumulative features of their elements, the local and global features of the visually appreciated form.18 How this works for physical objects is shown in Fig. 10.7 (left). Readers might see a cube in the combination of features, though the lines are not actually there. Local features are the circles and cutouts, global features the relations between them. Recognizing words is similar (Fig. 10.7, right): The words are likely to be read as ‘the cat’, though the identical characters in the middle of both are neither H nor A. Meaning is derived from the characters themselves, the context of adjacent characters, and learned associations between written forms and language. Trained object recognition and learned lexical associations likely account for the wide variability in script forms—Chinese to Arabic—as well as the flexibility with which graphic elements are able to map to different levels of language: logographic, syllabic, and alphabetic scripts. The potential for a written object to be recognized through its features and associated lexically appears independent of its actual form, as long as that form has certain characteristics: It must be small and simple enough that its reproduction involves little time or effort. Its meaning must be socially shared, not idiosyncratic. And it must be written by hand and produced at a volume sufficient to garner training effects.

Fig. 10.7. Feature recognition of physical and written objects. (Left) A cube is recognizable in the combinations of features: Local details are the circles and cutouts; global cues, the relations between them. (Right) The words ‘the cat’ are recognizable, although the middle characters are malformed and identical. Linguistic recoverability depends on local details, the characters themselves; global features, the context of adjacent characters; and learned lexical associations. Image created by the author. A version was previously published in Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’, p. 288, Fig. 2a.

Recognizing written objects through their features and associating them lexically may explain why written non-numerical signs became less depictive of the pictures and icons they had been originally, a change facilitating their productive recombination.19 18 Adelman, Marquis, and Sabatos-DeVito, ‘Letters in Words Are Read Simultaneously, Not in Left-to-Right Sequence’. 19 Roberts, Lewandowski, and Galantucci, ‘How Communication Changes When We Cannot Mime the World: Experimental Evidence for the Effect of Iconicity on Combinatoriality’.

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As hands and eyes were trained to produce characters, brains became trained to recognize the features of written characters and associate them lexically, relaxing the need for the characters to depict. That is, characters became recognized topologically, which allowed their shapes to deform. And in turn, this allowed them to become more alike, contrastive, and simpler. By ‘more alike’, I mean that later forms were closer in appearance to each other than the early forms had been. Consider pictographic and ideographic characters like the ones in Fig. 10.5 (a) and (c) and Fig. 10.6 before the Jemdet Nasr period. They are easy to tell apart and identify, not just because they resemble objects but because their differences are obvious. Contrast this with a body of cuneiform writing, like the characters in Fig. 10.6 from the Early Dynastic IIIa period and thereafter. Here the differences are more subtle, so the characters are more difficult to tell apart and identify. Visual discriminability and individuation—being able to tell characters apart and identify any particular character as itself—depend on familiarity. Essentially, unfamiliar objects are more difficult to tell apart, so the clues distinguishing them must be fairly obvious. With familiarity, objects become easier to tell apart, and the distinguishing clues can become subtler. This effect is also found in facial recognition, where it affects things like eyewitness identifications, particularly across ethnic groups.20 By ‘contrastive’, I mean that as characters lost their depictiveness, became more alike, and were discriminated and individuated by means of increasingly subtle clues, those clues converged on likenesses and differences. This is the process whereby graphic elements become a contrastive system, finding points of maximum difference within a fairly narrow range of variability. It occurs without change to the way graphic elements are mapped to language, so writing can become a contrastive system whether it is logographic, syllabic, alphabetic, or some combination. In the logographic/logosyllabic cuneiform script, characters were increasingly distinguished through the presence or absence of particular wedges, the angles at which they were made, or their number. Alphabetic analogies include directionality, differentiating b from d, p, and q; length, distinguishing i from j; and the presence or absence of elements, the differences between c, e, and o, as well as h, l, m, n, and r. This change also shows why scribal training became more formal. Telling the initial pictographic and ideographic characters apart and approximating their meanings through resemblance had been relatively easy. But as characters became more alike, the differences between them would become too subtle for anyone who lacked the requisite familiarity, necessitating training. And by ‘simpler’, I mean that characters lost some of their detail toward the Old Babylonian and Middle Assyrian periods. This suggests there was an optimization or balance of local and global detail. Local detail helps novices but slows proficient readers, who make greater use of global cues. In modern scripts, this effect is found in dia-

20 Brigham et al., ‘The Influence of Race on Eyewitness Memory’; Hayward et al., ‘The Other-Race Effect in Face Learning: Using Naturalistic Images to Investigate Face Ethnicity Effects in a Learning Paradigm’.

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critics for tones in African languages21 and vowels in Hebrew.22 In these modern cases, discussion involves the potential for ending up with different forms of writing, one for novices and one for proficient readers, with the trade-off that readers must learn both. In Mesopotamian writing, local details were increasingly omitted in later writing periods, suggesting readers were making greater use of global cues. It’s also true that the more detail a character has, the longer it takes to write, so character simplification enabled the speed of production to increase. Handwriting involves motor movements that are planned and executed by Exner’s Area of the brain. Exner’s Area is thought to help us recognize the gesture in the written word;23 the idea is that when we see a written form, we mentally simulate the physical gestures involved in producing it. Activity in Exner’s Area is particular to handwriting, not just any fine work involving the hand, so while it’s possible to produce characters, say, by carving them in stone, carving differs from writing in both the movements used and the character repetition involved. Writing by hand is known to have specific effects: It improves fine motor skills, hand–eye coordination, recognition and recall functions, lexical retrieval, and tolerance for ambiguity in how characters are formed.24 In the emergence of literacy, handwriting would have been critical. Not only would it have yielded these same benefits, it also afforded a mechanism for the continual adjustment of the material form, something essential to change in the cognitive system. Change in how the characters were written implies things like standardization, automaticity, and tolerance for ambiguity in character form. Standardization is forming each character with particular strokes in a particular order. When writing first began, there was no such protocol. Standardization emerged gradually; it’s seen in the strokes used and the order in which they were made, things that became increasingly codified.25 Over time, these show that production becoming standardized, and standardization shows handwriting becoming automatic. Automaticity frees up cognitive resources like attention and working memory. The same thing occurs when we learn to drive.26 At 21 Bird, ‘When Marking Tone Reduces Fluency: An Orthography Experiment in Cameroon’; Koffi, ‘Towards an Optimal Representation of Tones in the Orthographies of African Languages’. 22 Ravid and Haimowitz, ‘The Vowel Path: Learning about Vowel Representation in Written Hebrew’. 23 Konnikova, ‘What’s Lost as Handwriting Fades’. 24 Giovanni, ‘Order of Strokes Writing as a Cue for Retrieval in Reading Chinese Characters’; James and Engelhardt, ‘The Effects of Handwriting Experience on Functional Brain Development in Pre-Literate Children’; Longcamp, Zerbato-Poudou, and Velay, ‘The Influence of Writing Practice on Letter Recognition in Preschool Children: A Comparison between Handwriting and Typing’; Sülzenbrück et al., ‘The Death of Handwriting: Secondary Effects of Frequent Computer Use on Basic Motor Skills’. 25 Bramanti, ‘Rethinking the Writing Space: Anatomy of Some Early Dynastic Signs’; Taylor, ‘Wedge Order in Cuneiform: A Preliminary Survey’. 26 Charlton and Starkey, ‘Driving without Awareness: The Effects of Practice and Automaticity on Attention and Driving’.

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first, we pay lots of attention to operating the car and conditions on the road. As proficiency is gained, we attend less overtly to these things, becoming alert only when conditions change. This leaves us free to attend to thoughts and conversations. In writing, automaticity lets us focus on what we’re writing, its content, rather than how, its production.27 This would have transformed writing from a tool that merely recorded mental content to one allowing its users to engage that content directly. When we write by hand, we develop a tolerance for ambiguity in how characters are formed. This lets us recognize them even if they are not particularly well formed, while providing writing an important mechanism for change. This tolerance is one of the things educators fear will be lost as we type more and handwrite less,28 though arguably, the loss will be offset by the standardized appearance of characters on screens, as well as the things computers are really good at that handwriting isn’t: speed of production, networking, information on demand, and facial expressions ჉. Being able to recognize characters even when they’re ambiguous is an essential element of cursive, script that trades accuracy of form for speed of production. Faster writing keeps up better with the speed of thought, so that writing can become a highly interactive engagement between mental content, material form, and behaviors interfacing the two. The materiality of writing—clay surfaces and styli—also influenced how characters were made. Making lines or impressions on clay produces furrows, which is how their order of production can be determined. Furrows also meant characters had to be relatively simple and made with deliberation, since complex characters and characters made in haste quickly become ‘an undifferentiated confusion of superimposed impressions’.29 In turn, simplicity improved speed of production, while deliberation improved legibility and reduced error. Simplicity would also have influenced production at a fundamental level. Characters taking hours or even minutes to make wouldn’t have recorded or communicated information efficiently because of their restricted volume, nor would they have supported the recombination needed to produce new signs. Over some 15 centuries, the cognitive system changed from a functional reading and writing, in which language was not very fluent, to a state more akin to the literacy we enjoy, where the material form represents language with fidelity (Fig. 10.8). Characters lost their depictiveness and then simplified, suggesting written objects were being recognized by their features and becoming a system of contrastive elements with an optimized balance of local details and global cues. Character production converged on wedges, improving legibility, visual discriminability, and individuation, and wedgemaking order became standardized, suggesting automaticity. By 2000 BCE, a literacy on par with what we mean by the term appears to have developed. Words were no longer being split between lines of text,30 a preference for integrity of form that would support object recognition. A ‘cursive script, with abbreviated signs, crowded writing, and un-

Tucha, Tucha, and Lange, ‘Graphonomics, Automaticity and Handwriting Assessment’. Konnikova, ‘What’s Lost as Handwriting Fades’. 29 Cammarosano, ‘The Cuneiform Stylus’, p. 79. 30 Cooper, ‘Sumerian and Akkadian’, p. 45. 27 28

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clear sign boundaries’ developed,31 demonstrating a tolerance for ambiguity that enabled an even greater speed of production. Experimentation was widespread, with writing applied to many new purposes,32 a concern with content suggesting automaticity had repurposed cognitive resources like attention. And training became highly formal33 because script could no longer be read without it. The types and rate of change also decreased around this time. Discussions of writing generally focus on its ability to subject language and ideas to analysis and communicate them across space and time.34 Certainly, the ability to write astronomical data and ideas made the movements of stars and other celestial bodies tractable to pattern detection, categorization, and conceptualization in a way that far exceeded what had been possible without the recording, rearranging, and refinement that writing made possible.35 Discussions also focus on writing as an invention, seeing its subsequent change as further intention and deliberation. Clearly, some subsequent change falls into this category, as for example, the various writing reforms that standardized character forms or reorganized the way they were to be written. But for a moment, consider a set of characters only as a material form. It is difficult to imagine what sort of thing literacy could be without this form. Interacting with it engaged specific behaviors and psychological responses in its writers. What each generation interacted with was much the same form previous generations had used. But their interaction, plus the malleability of the written form, enabled each new generation to realize incremental change. The process was not teleological because no such goal as script nor guiding purpose of achieving literacy was possible. Change would have been largely imperceptible, except in hindsight. Writing changed simply because people used it. After centuries of communal use and adjustment, the material form was highly adept at eliciting specific responses. And yet while it changed, the material form remained aligned to common behavioral and psychological capacities because of the many hands shaping it, a process of leveling in which highs and lows counteracted each other and kept the form synchronized to the average. The material form embodied and made available the incremental change in behaviors and psychological responses realized and accumulated by past generations. It acted as a medium for recreating those changes in new individuals, as they learned to interact with it. And through characteristics and mechanisms like malleability and contrasts of form and structure, it afforded possibilities for future change.

Veldhuis, ‘Levels of Literacy’, p. 72. Veldhuis, ‘Cuneiform: Changes and Developments’, p. 3. 33 Veldhuis, History of the Cuneiform Lexical Tradition. 34 Olson, The World on Paper: The Conceptual and Cognitive Implications of Writing and Reading. 35 Watson and Horowitz, Writing Science before the Greeks: A Naturalistic Analysis of the Babylonian Astronomical Treatise MUL.APIN. 31 32

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Fig. 10.8. The development of literacy from writing. Mesopotamian writing changed dramatically, especially before 2000 BCE. These changes were sorted into seven categories: lexicography, dictionary-like compilations of words; organization, the layout of words on writing surfaces; syntax, how characters, words, and phrases are arranged to reflect language; orthography, conventionalizations of signs and sign combinations; applications, the purposes to which writing and scripts are applied; curriculum, the systemization of training for the writing system; and language, the degree to which the writing system expresses an identifiable language. An earlier version of this graph appeared in Overmann, ‘Beyond Writing: The Development of Literacy in the Ancient Near East’, p. 297, Fig. 9. The data were sourced from Bramanti, ‘Rethinking the Writing Space: Anatomy of Some Early Dynastic Signs’; Cooper, ‘Sumerian and Akkadian’; ‘Babylonian Beginnings: The Origin of the Cuneiform Writing System in Comparative Perspective’; Englund, ‘Texts from the Late Uruk Period’; Hyman, ‘Of Glyphs and Glottography’; Krispijn, ‘Writing Semitic with Cuneiform Script: The Interaction of Sumerian and Akkadian Orthography in the Second Half of the Third Millennium BC’; Schmandt-Besserat, How Writing Came About; Taylor, ‘Tablets as Artefacts, Scribes as Artisans’; ‘Wedge Order in Cuneiform: A Preliminary Survey’; Veldhuis, ‘Cuneiform: Changes and Developments’; History of the Cuneiform Lexical Tradition; ‘Levels of Literacy’.

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THE EFFECTS OF WRITING ON NUMBERS There are important differences between literacy, the ability to read and write, and numeracy, the ability to reason with numbers. Some of these differences have already been noted, like the representational modes and associated ambiguity. Literacy involves an initial repertoire of signs with conventionalized ‘this-means-that’ associations. These must be written by hand, enough hours per day and days over time, to train object recognition and afford automaticity. Signs must be simple, and the material form malleable enough, to enable production, repetition, recombination, and change. And signs must not be numerical, as numbers lack qualities needed to drive the written form toward the fidelity to language so essential to literacy. Numbers, on the other hand, can be represented with non-written forms like tallies and tokens. And even when they are written, numbers don’t need the phonetic specification indispensable to linguistic fidelity in non-numerical writing. In fact, including a phonetic component increases the visual complexity of characters: seven instead of 7. There are situations in which this complexity might be advantageous, like reducing the possibility the represented value might be tampered with. However, in general, phonography only degrades the ability of numbers to represent spatial, topological, and geometric relations, qualities essential to pattern recognition and information manipulation. Beyond the fact they don’t need to be written, numbers are only a subset of the lexicon of any language, and they can be represented with just a few signs and without phonetic value. This means a system of numerical notations is incapable of supporting the phenomenon of literacy, which involves representing more of the lexicon than just numbers. This in turn requires more signs than numbers need, as well as the ability to represent sound values to specific the words intended. And while both numbers and language have structure and rules that distinguish well-formed statements from illformed and nonsensical ones, the two are organized quite differently.36 For example, there are no restrictions on rearranging numbers to form new ones; the permutations possible with 1, 2, and 3 are all well-formed and meaningful: 123, 132, 213, 231, 312, and 321. This is not true of non-numerical language, which cannot be rearranged with a similar retention of meaningfulness and well-formedness: John loves Mary differs from Mary loves John in ways beyond simple order, and *John Mary loves, *loves John Mary, *loves Mary John, and *Mary John loves are neither well-formed nor particularly meaningful. This distinction can be made at other levels of language as well: cat, *cta, act, *atc, *tca, and tac are differentially meaningful or nonsensical. And this gets at another important point: No matter how they are rearranged or how often written, numbers cannot suggest features of language like word order, another reason they cannot give rise to literacy. There are neuroanatomical differences as well. As I mentioned in Chapter 4, different parts of the brain support language and numeracy, and the neuroanatomic structures supporting these functions are separate enough to be doubly dissociable. Within the neuroanatomy of reading, there are subtle differences that differentiate reading numbers from reading language: Distinct regions identify written numbers and non36 Chrisomalis, Numerical Notation; Comrie, Language Universals and Linguistic Typology; Greenberg, ‘Generalizations about Numeral Systems’.

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numerical characters,37 and the regions active in numeracy and language connect to different parts of Exner’s Area, the region implicated in the movements of handwriting.38 Admittedly, the importance of these differences is unclear. However, there appear to be functional differences as well. The visual appreciability of numbers improves when they are separated by spacing or punctuation, something that makes nonnumerical writing more difficult to read. Transpositions and substitutions in letter and word arrangements might go unnoticed because of error-correction functionality in reading non-numerical language, the phenomenon making it difficult to spot mistakes in proofreading; reading speed also decreases as the number of errors increases.39 These things have no parallels in numbers, where transpositions, substitutions, and the automatic filling analogous to informed guessing yield incorrect quantities, not corrections recovering intended content. Given these differences, it is not surprising the two are dissociable phenomena: There are quite a few systems of numerical notations without non-numerical writing, and a few systems of writing without numbers.40 And it’s probably no accident the societies inventing both—Mesopotamia, Egypt, China, and Mesoamerica41—were also the ancient world’s great mathematical traditions. When the two co-occur, writing has several interesting effects on numbers, and these arguably yield the qualities that make numbers the concepts we know them as today. In short, writing, and not just writing for numbers but writing for non-numerical language as well, appears essential to elaborating a system of numbers as a complex mathematical system. First, numbers that aren’t written, like the seven small cones representing the number seven in the tokens, early numerical impressions, and proto-cuneiform notations, are collections of discrete objects. In contrast, a handwritten number, like the cuneiform seven (ൢ), becomes recognized through the neurological training effects associated with handwriting. As written objects, such signs were more likely to have been conceived as coherent entities than collections, even though they too consisted of multiple elements. I don’t suggest unwritten numbers—groups of tokens, for example— are always and only conceptualized as collections and never as entities. Certainly, language might influence numbers toward entitivity, since it’s possible to say ‘there are seven’ in a way that seven doesn’t seem purely adjectival. However, we learn to recognize handwritten signs like we do physical objects. Written numbers, though still comprised of multiple elements the way their unwritten instantiations were, would have become coherent objects with associated lexical meanings. The concepts associated with written signs would have been influenced toward becoming entities in their own right, rather than remaining the collections of discrete objects their predecessors were. 37 Grotheer, Ambrus, and Kovács, ‘Causal Evidence of the Involvement of the Number Form Area in the Visual Detection of Numbers and Letters’. 38 Klein et al., ‘Differing Connectivity of Exner’s Area for Numbers and Letters’. 39 Rayner et al., ‘Raeding Wrods with Jubmled Lettres: There Is a Cost’. 40 Chrisomalis, Numerical Notation. 41 Powell, Writing: Theory and History of the Technology of Civilization; Senner, ‘Theories and Myths on the Origins of Writing: A Historical Overview’.

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Functionally, written numbers were concise to a degree their precursors were not. Concision let relations like multiplication and reciprocation be collected into tables, presenting relational information at a volume that had eluded even the protocuneiform signs: length and area measures (which emerged 2999–2000 BCE), reciprocals (2050–1500 BCE; also see Fig. 10.9), and powers of the integers (2000–1600 BCE).42 Scribes learned these tables as part of their training, something known through the schoolhouse texts they left behind them.43 Consistent with the idea numerical relations were being explicated, recorded, codified, and learned through tables, an increased awareness of numerical relations is suggested by a writing reform implemented in the Ur III period (2100–2000 BCE) that altered how large numbers were written: ‘ŠÁR × 10 repeated two, three, and four times’ became ‘ŠÁR × 20, 30, and 40’.44 The increased accessibility of numerical relations made possible by writing and tables, along with the mandate to recreate and learn the tables as part of scribal training as it became increasingly formalized, would have helped facilitate the conception of numbers as objects related numerically to other, similar objects. This awareness would redefine numbers through their relations to each other, making them entities in a relational system. These changes gave scribes new options in calculating. Certainly, they could still use tokens, whose manipulability made them suitable for this purpose in a way the fixed notations were not. But now scribes could also recall relations from memory or look them up in tables. When added to the ability to record interim steps that writing made possible, these would enable calculations to become more complex, and they also helped shift numeracy toward mental knowledge even further than was true of the earlier tokens. And just being able to write the results of calculations would have enabled scribes to visualize and apprehend relations between wholes and parts in a way not possible with earlier technologies, since rearranging tokens into parts destroys the whole and tally notches cannot be rearranged. Concision plus the stability writing provided also allowed attributes like numerical base to be made more explicit and organized in ways that would begin to crystallize as a system of place value by the Old Akkadian period (2350–2250 BCE).45 The idea tokens continued to be used for calculating is supported by scribal errors that give the impression of tokens being comingled,46 as well as limitations in the number of places to which a calculation could be carried out.47 Their continued use also makes sense because notations are fixed. We tend not to think of notations in this way; their fixedness is obscured by our use of interactive strategies that cross out and reRobson, ‘Mesopotamian Mathematics’; Mathematics in Ancient Iraq: A Social History. Postgate, Bronze Age Bureaucracy: Writing and the Practice of Government in Assyria; Proust et al., Tablettes Mathématiques de Nippur; Robson, ‘Tables and Tabular Formatting in Sumer, Babylonia, and Assyria, 2500 BCE–50 CE’. 44 Powell, Sumerian Numeration and Metrology, pp. 75–77. 45 Robson, ‘Mesopotamian Mathematics’. 46 Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin, p. 2. 47 Friberg, ‘Methods and Traditions of Babylonian Mathematics: Plimpton 322, Pythagorean Triples, and the Babylonian Triangle Parameter Equations’. 42 43

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write them, as well as algorithms that decompose complex problems into series of small mental judgments whose output values form the basis for crossing out and rewriting input values. Fixedness meant notations were suitable only for recording, at least until the requisite relations and algorithms for manipulating relations were developed and the preference for using mental knowledge rather than physical manipulation had taken hold. These developments would continue over the next several millennia, as evidenced by discussions in medieval Europe regarding the respective merits of abaci and algorithms.48 Of course, this view of how numbers develop historically requires understanding them as concepts that don’t come preloaded with relations and algorithms, properties that must be painstakingly worked out. There must also be a mechanism for socially sharing the knowledge, and that mechanism cannot be language because language indiscriminately labels numbers alike, whether they have many relations or few. That is, there’s no way to determine from a word meaning three whether it is three the subitized appreciation, three the ordinally related number, or three the relational entity equivalent to five minus two and square root of nine. The ability to write non-numerical language was equally crucial to mathematical elaboration. As writing became more expressive, it was used to record calculations, not as results, as had been true with earlier technologies like tokens, and not as equations, like we would do. Instead, calculations were recorded as narrative description. It is interesting to note that although cuneiform was logographic and logograms express words, arithmetical operations were not expressed in succinct characters analogous to our plus and minus signs. In fact, it would take another several millennia to formulate and conventionalize concise semasiographic notations for even the simple arithmetical operations49 and free operational signs and calculations from the ‘slow and hesitant’50 expressiveness of language.

48 Reynolds, ‘The Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators’; Stone, ‘Abacists versus Algorists’. 49 Schulte, Writing the History of Mathematical Notation: 1483–1700. 50 Sfard and Linchevski, ‘The Gains and the Pitfalls of Reification? The Case of Algebra’, p. 198.

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Fig. 10.9. Mathematical tablet (Erm 14645). (Left) Obverse face of a mathematical tablet in the Sumerian language, provenience uncertain, dated to the Old Babylonian period and held in the collection of the State Hermitage Museum, St. Petersburg, Russian Federation. The tablet is a table of reciprocals, numerical relations comprising a functional multiplication and division. For 60, the reciprocal of 2 is 30 because 2 × 30 and 30 × 2 equal 60, implying 60 ÷ 2 = 30 and 60 ÷ 30 = 2. The notation for 18 is ‘20 la2 2’, or ‘20 minus 2’.51 This is an instance of ‘backward counting’,52 where a productive magnitude decreases by a subitizable amount; it highlights the fact that Mesopotamian numbers were formed in ways typical across number systems generally. (Right) The reciprocals as translated from base 60 into base 10 in modern numbers. Image of mathematical tablet courtesy of The State Hermitage Museum, St. Petersburg, Russia.

These narratives were ‘largely utilitarian and very much of a “cook-book” variety. (“Do such and such to a number and you will get the answer.”)’.53 The steps and vocabulary they used evoke the manipulation of physical tokens. There were two types of addition and subtraction, differentiated by whether or not addends, minuends, and subtrahends remained identifiable once commingled or separated; there were four multiplications, including one—reciprocals (Fig. 10.9)—suggesting tokens were being rearranged as different groups.54 These operations suggest the Mesopotamian concept of number differed from ours today, in more ways than its sexagesimal organization and cunei51 Friberg, ‘A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma’, p. 14. 52 Menninger, Number Words and Number Symbols: A Cultural History of Numbers, p. 16. 53 Devlin, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe, p. 1. 54 Høyrup, Lengths, Widths, Surfaces, p. 19.

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form notation: Numbers that sometimes keep their identity and sometimes do not when manipulated arithmetically, like the Mesopotamian numbers did, differ fundamentally from ones making no such distinction. Four doesn’t occasionally disappear when it is added or multiplied today. The overt presence of materiality in Mesopotamian calculation also offers a different argument against the ‘concrete–abstract’ distinction. By the Old Babylonian period, Mesopotamian mathematics had reached the height of its elaboration:55 arithmetic progressions; multiplication and division by reciprocals; powers of the integers; squares, inverse squares, diagonals of squares, and cubes; properties of lengths and areas; quadratic equations for squares, circles, circumferences, areas, diameters, right triangles, and rectangles; and a type of algebra enabling the manipulation of areas and lines to find unknown sides of squares.56 Mathematicians had also established the sexagesimal place value system as a way of integrating the different metrological systems.57 These developments followed the separation of conjoined quantity and commodity by millennia, and they exemplified the state when Mesopotamian numbers were presumably their most abstract. Yet as the overt materiality of their calculations show, Mesopotamian numbers still very much depended on the material forms used to represent and manipulate them. I’m not suggesting that using material forms determines a quality of abstractness or concreteness. Rather, I believe the use of materiality may be less visible when the material forms and the operations conducted on them are familiar and the concepts have become distributed over multiple material forms, making them irreducible to any particular form. Material dependence may be more apparent in Mesopotamian numbers simply because our knowledge of them is more limited, though arguably, the two systems have more in common than not in their use of material forms.

MORE EFFECTS OF WRITING ON NUMBERS Being written had helped numbers become entities in a relational system. But numbers changed even more from being written: They acquired place value, redistributing numerical information that had previously been explicitly encoded in the material form as implicit knowledge that had to be supplied by the user.58 Forms requiring counting would also ultimately become encoded as signs with non-countable elements, a further shift toward implicit knowledge.

55 Høyrup, ‘Written Mathematical Traditions in Ancient Mesopotamia: Knowledge, Ignorance, and Reasonable Guesses’; Robson, Mathematics in Ancient Iraq. 56 Robson, ‘Mesopotamian Mathematics’. 57 Høyrup, In Measure, Number, and Weight: Studies in Mathematics and Culture; Lewy, ‘Origin and Development of the Sexagesimal System of Numeration’; Seidenberg, ‘The Sixty System of Sumer’. 58 Zhang and Norman, ‘A Representational Analysis of Numeration Systems’.

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Systems of numerical notations can be organized along two dimensions.59 The first, intra-exponential organization, describes how numerical signs are constituted and combined within each power of the base. In Stephen Chrisomalis’ typology,60 there are three major types. In cumulative systems, signs are repeated with the expectation they’ll be added, so III in Roman numerals means three because each sign means one and there are three of them. In ciphered systems like the familiar Hindu–Arabic numerals, a single sign is used for each power of the base: 3, a single sign, rather than the tripartite III. Multiplicative systems like Chinese numerals combine a numerical sign with one specifying the power of the base, so 30 would be expressed as ୕༑, where ୕ is the sign for 3 and ༑ is the sign for tens. The second dimension, inter-exponential organization, describes how the values of the signs for each power of the base are combined to realize the total value of a numeral phrase. Here there are two major types. Additive systems sum the values to realize total value, so XXVIII in Roman numerals means 28 because two tens, one five, and three ones yield this number when summed. In positional systems, place implies value, so a 3 might mean three, thirty, or three hundred, depending on whether it was placed in the ones, tens, or hundreds column. Tallies had been one-dimensional, accumulating without grouping. Fingers can be used to accumulate and are naturally grouped by fives and tens, a nascent twodimensionality underlying the expression of explicit two-dimensionality in subsequent technologies. The Neolithic tokens, early numerical impressions, and proto-cuneiform notations were unambiguously two-dimensional, not only accumulating but grouping as well. Accordingly, they are classified as cumulative–additive systems, cumulative because they repeated signs for particular values with the expectation these would be added together, and additive because the values of all the signs in a specific collection were summed to realize its total value. Babylonian cuneiform, on the other hand, was a cumulative–positional system, cumulative because wedges (൫ and ೈ) were added together and positional because successive numbers were understood to represent sequential powers of the sexagesimal base. Cumulative–positional systems like Babylonian cuneiform develop from cumulative–additive systems like tokens and proto-cuneiform.61 For Mesopotamian numbers, the organization of tokens by the magnitude of their value had influenced the use of this organization in early numerical impressions. In the proto-cuneiform notations, the subsequent elaboration of numerical ordering took several forms. While elements of individual signs were still ordered by the magnitude of their values, multiple signs could now be represented by impressing them onto the obverse or reverse faces or the tops, bottoms, or sides of an artifact. Multiple signs could also be distinguished through placement within different cells or adjacent columns on a single face. This organization, especially as the volume of information expanded, the use of tables increased, and

59 Chrisomalis, Numerical Notation, pp. 11–13; also see Widom and Schlimm, ‘Methodological Reflections on Typologies for Numerical Notations’. 60 Chrisomalis, p. 13. 61 Chrisomalis, pp. 381–385.

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cellular/columnar organizations became more complex, would have been relatively easy to simplify as a place-value system. Value understood positionally (10) rather than being explicitly indicated through encoding (X) or the presence of an extra sign (୍༑) depends on user knowledge. Admittedly, none of these examples is completely explicit: Just as users must know the value in the tens column is multiplied by 10, they must also know the ciphered form X has the value 10 and that the signs for 1 and 10 are to be multiplied. Arguably, numbers always involve some implicit knowledge, as neither material forms nor language explicitly represent properties like relations. The implicit–explicit mix occurs early in numbers, since ordinality implies numbers later in the sequence are bigger than earlier ones, and the number sense enables judgements of relative size well before numbers are conceptualized, ordered, and related. As relations become more defined—not just bigger or higher than but one of these means ten of those and six more than—the amount of knowledge users must supply increases. While knowledge differs from the properties of material structure and organization that persist as habits and expectations, it too falls within the mental input space of conceptual blending, discussed in Chapter 3. Implicit knowledge enables whatever the material form does explicitly represent to become even more concise, further enhancing its ability to make patterns appreciable and information manipulable. Encipherment also represents a shift to implicit knowledge. How cumulative notations like ൫൫൫൫൫൫൫ become ciphered ones like 7, and whether or not they will, are plausibly related to our perceptual experience of quantity. Subitizable quantities are appreciable, so numerical signs with one, two, or three elements need little alteration: ൫, ൬, and ൭. Because they instantiate quantity in the range we can see, these forms have tended toward conservation, even when handwriting provided them a freedom to vary that fingers, tallies, tokens, and impressions had not. In fact, like fingers and tallies, written signs for one, two, and three have taken form as one, two, and three strokes more often than not, something that can be verified by comparing notational forms across systems. These forms also persist for millennia, verifiable by examining within-system change over time (Fig. 10.10). The familiar Hindu–Arabic number sign for 1 remains the single stroke it has been, not just since writing began but in precursor technologies as well. Even the number signs for 2 and 3 are still basically two and three strokes each, handwritten quickly with a diagonal tilt from the upper right to the lower left. Conservation of form in written subitizable numbers is even more apparent in Chinese, where one, two, and three remain the horizontal strokes they were in even the earliest forms known for this writing system.62

62 Ager, ‘Omniglot: The Online Encyclopedia of Writing Systems and Languages’; Branner, ‘China: Writing System’; Martzloff, A History of Chinese Mathematics.

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Fig. 10.10. Change in numerical and non-numerical signs. The signs shown on each row are related to each other, although as represented in the figure, the lineages are incomplete. Written representations for numbers and language experience distinct neurofunctional pressures that influence the forms they take, how these forms transmit across languages and cultures, and how they change over time. Signs for numbers instantiate, so signs for subitizable numbers conserve. Signs for large numbers simplify, which makes them increasingly subject to processes and mechanisms of writing change. Signs for nonnumerical language signify, so they must specify meanings and sounds, which alters their visual appearance and subjects them to the greatest amount of change. Information compiled and images adapted from Chrisomalis, Numerical Notation: A Comparative History; Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer; Nissen, ‘The Archaic Texts from Uruk’; Tompack, A Comparative Semitic Lexicon of the Phoenician and Punic Languages; and data from the Cuneiform Digital Library.

When large numbers—those higher than about four—are written cumulatively, they contain a number of elements whose quantity is not subitizable. Instead, these elements must be counted, an awkwardness influencing their rearrangement into subgroups of smaller, more appreciable quantities. This organization emerged in the columnarization of early impressions, and it became increasingly formal in the handwritten characters for large numbers: ඔ, ඕ, or ඓ instead of ൫൫൫൫; ඘, ඙, or ඗ instead of ൫൫൫൫൫൫൫; ග or ඝ instead of ൫൫൫൫൫൫൫൫൫. Even as subitizable subgroups, these signs would have remained under some pressure for being counted, making it likely they would simplify further as ciphered forms that didn’t involve counting. And indeed, ciphered forms like 4, 7, and 9 would ultimately develop, but well after the cuneiform era had ended. Once ciphered, these signs would become subject to the processes and mechanisms that change written forms, like topological recognition and visual distinguishability. Quantities so high that counting was infeasible had been bundled and represented conventionally much earlier: ೈ and ౶ were used in tokens, impressions, and proto-cuneiform before they appeared as handwritten cuneiform signs. Production by means of handwriting made these ciphered forms subject to the processes and mechanisms of written change.

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Whether numbers are large or small, ciphered or cumulative, their semasiography means their written forms are transmitted across time and adapted to different languages and cultures without phonetic change and generally without semantic change. Since semasiographic forms don’t specify phonetic values, phonetic adaptations aren’t even possible. Semantic adaptation, though infrequent, is possible when number systems are organized differently; an example is the Elamite adaptation of Sumerian sexagesimal number signs to represent their indigenous decimal numbers. Semasiography limits the amount of change experienced by written numerical signs, as can be seen by comparing the cumulative–additive numerical impressions of the Uruk V period to the cumulative–positional cuneiform numbers of the Old Babylonian period in Fig. 10.11.

Fig. 10.11. Change in numerical signs. Both artifacts contain signs with repeated elements whose meaning and total value was derived by accumulating their quantity, though the earlier system was cumulative–additive, the later cumulative–positional. Compared to the amount of change seen in non-numerical signs across the same span of time (Figs 10.6 and 10.11), the forms of numerical signs are relatively consistent over time. (Left) Administrative tablet (W 6245,c) from the city of Uruk, assigned to the Uruk V period and held in the collection of the Vorderasiatisches Museum, Berlin. Its columns of N14 and N01 signs are impressed repetitions, suggesting they would have been conceptualized as collections, much as tokens were. (Right) Mathematical tablet (UM 55-21-356) from Nippur dated to the Old Babylonian period and held in the collection of the University of Pennsylvania Museum of Archaeology and Anthropology. The tablet is a list of squares; the sixth line is the first one with an unbroken initial number and reads ‘36 is the square of 6’.63 The signs are handwritten, suggesting the numbers would have been conceptualized as entities. Image of administrative tablet © Staatliche Museen zu Berlin – Vorderasiatisches Museum, Foto: Olaf M. Teßmer. Image of mathematical tablet courtesy of the Penn Museum, object UM 55-21-356 obverse. 63 Robson, ‘More than Metrology: Mathematics Education in an Old Babylonian Scribal School’, p. 360.

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In comparison, writing for non-numerical language undergoes a great deal of change (Figs 10.6 and 10.10). These signs are already under pressure to achieve greater semantic and phonetic specificity, achieved by altering their visual appearance through techniques like phonography that specify words. Relative to numerical signs, greater deformation is possible in non-numerical signs because their topological recognition is unconstrained by processes or mechanisms analogous to quantity perception and instantiation. Adaptation to different languages or cultures keeps non-numerical writing under pressure to specify meanings and sounds. When a system of writing is adapted, it initially does a poor job of representing sounds and meanings by an amount that varies with the linguistic and cultural affinities between the two populations. This means its visual appearance must undergo whatever additional change is needed to specify new sounds and meanings to an acceptable level. Now that we’ve explored how early Mesopotamian writing became a script capable of supporting literacy and what this had to do with elaborating numbers as a relational system, the next chapter looks more closely at separating the representations of quantity and commodity. This change in material form not only facilitated the development of the Mesopotamian literary and mathematical traditions, it also has the potential to illuminate aspects of human cognition more broadly.

CHAPTER 11. THE ROLE OF MATERIALITY IN NUMERICAL CONCEPTS When Spanish seafarers first visited Easter Island in 1770, one of the ship’s officers— possibly the navigator, Don Francisco Antonio de Agüera y Infanzón—compiled a list of words obtained from the islanders by touching body parts like arms and legs, pointing to objects like the large stone idols dotting the coastline, and pantomiming actions like standing up and eating.1 When it came to numbers, however, the Spaniards seem to have shown the islanders written signs for the numbers one through ten. Not unexpectedly, the words this elicited bore no resemblance to the Austronesian lexical numbers used throughout Polynesia, whose use on Easter Island would be documented a mere four years later by the English expedition led by Captain James Cook.2 Today there is consensus that ‘the Spaniards who compiled this list erroneously identified as numerals words which never were numerals’.3 The Spaniards were not wrong, however, in thinking their written forms meant numbers, only about what was involved in accessing their meaning. A material form mixing explicit representation with implicit knowledge requires the user have the requisite knowledge, and perhaps a neurological reorganization gained by interacting with the form enough to become proficient as well. If the Spaniards had perhaps displayed a tally instead, the islanders might have recognized its numerical function immediately. For one thing, they were known to have used such devices themselves.4 But a tally also represents numbers with a smaller implicit component. Notches lack, for example, the encipherment and positionality assumed in the notational forms the Spaniards used. Using a tally also doesn’t involve a neurological component analogous to learning to read. González de Haedo, The Voyage of Captain Don Felipe Gonzalez in the Ship of the Line San Lorenzo, with the Frigate Santa Rosalia in Company, to Easter Island in 1770–1771, Transcribed, Translated, and Edited by Bolton Glanvill Corney. 2 Cook, A Voyage towards the South Pole and Round the World, Performed in His Majesty’s Ships, the Resolution and Adventure, in the Years 1772, 1773, 1774 and 1775. Volume 2, Book III. From Ulietea to New Zealand. 3 Fedorova, ‘The Rapanui Language as a Source of Ethnohistorical Information’, p. 54. 4 Routledge, ‘Survey of the Village and Carved Rocks of Orongo, Easter Island, by the Mana Expedition’. 1

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What if the Spaniards had shown the islanders an abacus instead of written notations? Given the general Polynesian faculty with counting, the accumulative function of an abacus would likely have been readily apparent from its unciphered counters and an opportunity to manipulate them. If its place value were a bit difficult to grasp, the twodimensional organization of Polynesian numbers, used and likely originating in practices like finger-counting and the Tongan ceremonial counting described in Chapter 5, might soon have made this clear as well. There are pertinent differences, however, between counting, a practice documented throughout the region, and calculating, which involves a knowledge of numerical relations and algorithms for manipulating them. While calculating by means of moving counters on an abacus differs from calculating by means of relational judgments scribbled in notations in terms of the specific motions used and the knowledge of relations and algorithms required, both devices depend on the user having the requisite knowledge. How quickly the islanders might have been able to master using an abacus for calculating would have drawn on relations and algorithms realized through their own counting practices. These contained heuristics that could have served as a basis, as ten pairs were recognized as equivalent to twenty singles throughout Polynesia. The sequence of material forms used to represent and manipulate numbers in Mesopotamia suggests how a representational system develops the ability to assume implicit knowledge and neurologically reorganize its users. These things have no parallel in the perceptual experience of quantity: Like other species, we simply see what is, quantity-wise, within our perceptual limits, and this ability interacts with all of the objects in an environment. Realizing quantity is shared by groups of objects, however, requires picking out and relating groups with the particular quantity. And as quantity is only one of several properties the groups of objects have, salient but non-pertinent properties like color, size, value, and type must be ignored. Selecting objects to represent particular quantities, and collecting quantity representation onto a single material form like the hand, begin a process of consolidation that ultimately yields a highly concise material form with a large implicit component. It takes the fingers and most of the toes of four individuals to represent the number 75, but only a single tally with 75 notches on it. Seven Mesopotamian tokens, impressions, and proto-cuneiform notations further consolidated the representation of 75 with bundling, but required users to know that each large cone was equivalent to six small spheres and each small sphere meant ten cylinders. Handwritten notations consolidated representation even further by expanding the implicit component: Four cumulative–additive Roman numerals (LXXV), three cumulative–positional cuneiform numerals (ਸ਼ೈൠ), three multiplicative– additive Chinese numerals (୐༑஬), and two ciphered–positional Arabic–Hindu numerals (75) not only assume that users know encoding conventions, implied operations, and in some cases, positional weights, but that they can also read. As new devices are added to the sequence, higher numbers, numerical relations, and algorithms for manipulating relations are added to the cognitive system for numbers, increasing the amount of information in it. As information increases, each new device further consolidates the information represented explicitly by assuming users will supply the rest of the knowledge needed to understand its conventions. As these changes occur, the need for users to have familiarity with the form and to become proficient at using it increases too. Familiarity and proficiency are gained through interac-

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tion, which occasions whatever neurological reorganizations are implied by the specific motor movements involved in manipulating the form and the recognition and recall functions needed to understand its conventions. In Mesopotamia, the sequence of devices culminates in notational forms, which assume users will supply the greatest amount of knowledge, relative to all the other devices in the sequence, along with the neurological reorganization associated with learning to read. Where the Spaniards went wrong, then, was in expecting the islanders to have possessed the implicit knowledge and neurological reorganization needed to understand notational forms as numbers. The Spanish concept of number would also have been extensible over the kinds of material forms that had factored into its construction, devices like the tally and the abacus. From the perspective of people enculturated into highly elaborated number systems like the Western one, precursor technologies represent a subset of existing numerical knowledge, and are apprehended on that basis. The reverse, however, would not have been true. From the perspective of people enculturated into number systems relatively less elaborated, like Polynesian numbers in the 1700s, the numerical concepts represented would likely differ significantly or be absent entirely, providing little or none of the implicit component required to understand things like notational forms. This penultimate chapter looks at the sequence of material forms used for representing and manipulating numbers in the Ancient Near East. The internal coherence of the sequence is reviewed and used as the basis for some ideas about why we incorporate new material forms into the cognitive system for numbers and how this elaborates numerical concepts. The behavioral, psychological, and conceptual developments attending material change are discussed, as are the implications of distributing concepts like numbers over multiple material forms. Finally, the question of what it means to be abstract is addressed.

THE SEQUENCE OF MATERIAL FORMS USED FOR COUNTING Material forms have properties or affordances we exploit for numerical representation and manipulation. As many of the affordances provided by the devices used with Mesopotamian numbers have already been mentioned in detail, the internal consistency and implications of their chronological sequence are highlighted here (Fig. 11.1). As shown in Table 11.1, each material form used for counting provides certain capabilities and structure. Fingers, for example, are neurologically integrated with the ability to appreciate quantity; they are also readily available and easily bridge the internal mental and external physical experience of quantity. But each material form is also limited in some fashion. In terms of capacity, no matter how they are counted, there are only four fingers and a thumb, 14 joints and segments, or 33 tips, joints, and segments per hand, and some of these quantities are more manageable than others. Persistence-wise, the hands are not well suited for representing quantity, since sooner or later they will be needed for some other purpose. These limitations may eventually motivate the incorporation of a new material form, one selected on the basis of affordances it shares with the hand (Chapter 2), as well as capabilities that can address its shortfalls.

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Fig. 11.1. Chronology of material artifacts used in Mesopotamian numbers. Consistent with their neurological properties and ethnographic prevalence, fingers are sequenced first, though evidence of finger-counting in Mesopotamian languages would not emerge until well after writing was invented. There is archaeological evidence of possible tallies in the Levant during the Late Upper Paleolithic and possible tokens throughout the Mesopotamian Plain during the Neolithic. However, the numerical meaning of artifacts remains more-or-less ambiguous to modern eyes until notational forms emerged in the mid-tolate 4th millennium BCE. The image of a hand is from the public domain. The tally was originally published in Davis, ‘Incised Bones from the Mousterian of Kebara Cave (Mount Carmel) and the Aurignacian of Ha-Yonim Cave (Western Galilee), Israel’, Paléorient 2(1), p. 182, Fig. 3; it is republished with the kind permission of Paléorient. The tokens were adapted from Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’, p. 29, Fig. 24. The proto-cuneiform notations were redrawn from Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East, pp. 28–29, Fig. 28. The cuneiform number signs are Unicode characters.

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One-dimensional devices like tallies share with the hand the ability to represent and accumulate quantity, while providing capacity and persistence beyond what the hand can provide. However, along with these capabilities, tallies and other one-dimensional devices inject new limitations of their own: Tallies are not manipulable, though they have a desirable integrity of form. Collections of pebbles, while manipulable, lack integrity of form. The quantity of both becomes difficult to appreciate without counting when higher than about four. Devices may also bring capabilities beyond those mitigating a predecessor’s shortcomings, as for example, they can be separated from the vicinity of the objects being counted or the person counting them. To the degree permitted by their form, they may also have a greater capacity for modification, rearrangement, and elaboration than the body does. These technologies represent key transitions and conceptual change. Initial quantity judgments are equivalences between sets of objects: as many as the fingers on one hand. Using the fingers for counting gives numbers their basic structure and organization: linearity, stable order, and anatomical grouping. Tallies are collections related to the objects they enumerate; they also represent the transition to material culture, a medium more public, imitable, and collaborative than bodies and behaviors and even more capable of accumulating and transmitting knowledge between individuals and generations. With tokens, numbers become collections related not only to enumerated objects but to each other as well, representing the emergence of knowledge-based numeration. And with notations, because of the neurological reorganization associated with handwriting and the concision that enables the collection of relational information at increasingly larger volumes, numbers become conceptualized as entities defined by their relations. The material devices used to represent and manipulate numbers are the source of numerical properties like discreteness, linearity, manipulability, and dimensionality. These properties become numerical properties because they are implicit to the material forms that make numbers tangible5 and comprehensible.6 But no single material form has all of the properties numbers eventually possess, a quality that makes numbers independent of, and ultimately irreducible to, not just to any single form but to all the forms used.7 The recruitment of each new material form means new properties are potentially added to the cognitive system for numbers, and hence, to numbers as concepts.

5 Malafouris, ‘Grasping the Concept of Number: How Did the Sapient Mind Move beyond Approximation?’ 6 Frege, The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number. 7 Overmann, ‘Concepts and How They Get That Way’.

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Concept and Transition Artifact Distribut- - Equivalences ed exem- - Emergence of plars number concept Fingers - Equivalences - Imposition of basic structure

Affordances

- Iconic/indexical

-

- Linearity - Stable order

-

Tallies

- Collections

related to enumerated objects - Use of material culture Tokens

Written notations

- Collections

related to each other, as well as to enumerated objects - Emergence of knowledge-based numeration - Conceptualized as entities - Numbers defined by relations

New Capabilities

- Perceptible quantity - Judgment of quantity

-

-

similarity or dissimilarity Neurologically integrated with quantity perception Ready availability Used in manipulating objects being counted Bridge between psychological, behavioral, and material domains Linearity Stable order Accumulation

Linearity (imposed) Stable order (imposed) Accumulation More operations Persistence Greater capacity (higher numbers)

Linearity (imposed) Stable order (imposed) Many operations Even greater capacity Number specified separately from commodity - Grouped - Two-dimensional structure (imposed) - Even greater capacity

recreation

- Capacity - Persistence - Public

New Limitations Unstructured Limited to subitizing Ephemeral Limited capacity Ephemeral Limited manipulability - Commodity unspecified

-

- Commodity unspecified

- Visually

indiscriminable

- Grouped (more

- Not manipulable - One-dimensional - Loose (limited

- Two-dimensional - Manipulable (relations,

- Not concise - Not suitable for

- Commodity encoded

- Multivalent

-

- Fixed (not

discriminable) operations)

Integrity of form Concise Persistent Increased volumes of data Monovalent Handwritten (literacy effects) - Whole–part relations (with tokens) - Greater calculational complexity - Manipulation by conceptual relations

integrity)

recording

manipulable)

Table 11.1. Affordances and limitations of material artifacts used for Mesopotamian numbers. The material technologies used for counting in Mesopotamia are listed in chronological order. The interplay of affordances and limitations suggest the inclusion of newer forms is motivated by the limitations of older forms, and that newer forms will be selected on the basis of affordances shared with predecessors and capabilities addressing their shortfalls. Earlier versions of the table were published in Overmann, ‘Concepts and How They Get That Way’, p. 162, Table 1 and Overmann, ‘Updating the Abstract–Concrete Distinction in Ancient Near Eastern Numbers’, p. 13, Table 2.

Recruiting new material forms also enables some of the properties of older forms to pass from explicit representation into the domain of implicit knowledge. As forms like tallies are incorporated, the productive grouping of the fingers persists, but its source is no longer apparent. As manipulable forms like tokens are incorporated, the origin of properties like linearity, stable order, and magnitude ordering becomes unobvious. And once numbers become written and hence fixed, the source of the manipulability need-

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ed to develop relations and algorithms is no longer explicit. These properties pass between generations not only in the form of explicit attributes of the material devices used for numbers but as the knowledge needed to use the devices, acquired along with behaviors, habits, expectations, and neurological reorganizations by interacting with them. There are other reasons material structure persists. One is that the material component of numbers, considered across a geographic region and over time, is a patchwork of forms in which older and newer ones often coexist. Simply, not everyone gets the newest device right away, and others may resist using it. This would have been true in ancient times as well, when factors like distance could significantly impede the spread of ideas and technology. There is also a tendency to retain older forms for numbers, rather than replace them. Concurrent use is something we do ourselves: We use notations in multiple forms, and we also count on our fingers, use tallies and other one-dimensional structures in counting, and manipulate coins, which have a similar relational schema and a token-like bundling/debundling function. Because newer devices are selected for their shared similarities and are then used similarly because of habits and expectations, they are structurally similar to older devices. It’s reasonable to think concurrent use of similarly structured devices would tend to reinforce their structural similarities. Structure may persist for neurological reasons, like the neural interactions and topographical structure underlying and perpetuating finger-counting. And it can persist through learning and practice: Not only do learning and practice involve structured interactions with materiality, we also learn to use material forms in the same way our teachers did, something practice reinforces, and this tends to inhibit our using them in different ways. In psychology, this effect is called functional fixedness, a cognitive bias that limits us to using objects only in the way we have learned to use them 8 and which is independent of the specific form or degree of elaboration of material culture. 9 In numbers, functional fixedness not only keeps us using material forms in the same way we learned to use them, it also transfers learned functionality across material forms, influencing the way we use new forms even when those forms don’t physically require it, like linearity and tokens. Incorporating new material forms also has the effect of introducing contrasts between forms, like the manipulability of tokens would have contrasted with the physical fixedness of the tallies preceding them and the notations following them. Contrasts increase the likelihood that properties will be noticed, explicated, and exploited. This is because we are more apt to notice what differs, which in turn might make us think about the differing quality and what it is about it that makes it different. When a newer form doesn’t work like we expect or want it to, based on our experience of older forms, this non-cooperation might draw attention to its functionality or characteristics; affect whether, how much, or how we use it; or put it under pressure to change. And 8 Birch and Rabinowitz, ‘The Negative Effect of Previous Experience on Productive Thinking’; Duncker, On Problem-Solving. 9 German and Barrett, ‘Functional Fixedness in a Technologically Sparse Culture’.

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we must consider more than the contrasts between older and newer forms that motivate the selection of new forms and influence the type of form selected, as have been discussed. There are also interactions and contrasts between material forms and language to be considered. Consider traditional Oksapmin body-counting. Both of the forms used for counting—the sequence of positions on the body and the sequence of counting words—are ordinal,10 and neither is particularly manipulable. As a result, these numbers have few, if any, relations beyond those implicit to an ordinal sequence, so they are not readily adaptable to arithmetic tasks that involve manipulating relations. Subtracting 7 from 16 in Western numbers is a matter of knowing 7 from 6 carries the 1 and leaves 9 in the units column and 0 in the tens column, matters drawing on algorithms that decompose the problem into small judgments, relations between numbers like the 9 separating 7 from 16, and knowledge of ciphered meaning and place value. These algorithms, relations, and implicit component are absent in the Oksapmin system, where 7 and 16 are conveyed by the terms besa and tan-nata and their respective positions on the forearm and the other ear in the sequence of body parts. Asked to add and subtract besa from tan-nata by a Western observer, Oksapmin informants created the complex double enumeration strategy previously detailed in Chapter 3.11 Keeping simultaneous track of two ordinal sequences—one to accumulate the subtrahend, the other to decrement the minuend—implies an increased demand on cognitive resources like attention and working memory. Given social circumstances requiring frequent performance of the task, the cognitive demand would likely motivate finding ways to make it easier—like incorporating a manipulable form, one with an increased potential for illuminating and manipulating relations between numbers. As is, the affordances of the material and linguistic forms are reinforcing, and neither form is capable of generating relations on its own. These non-contrastive characteristics are most likely to yield stability, especially in numerical systems used by small, relatively isolated groups with less internal or external complexity to manage than larger groups in frequent contact. And given that body-counting systems span multiple language families across the isolating geography of Papua New Guinea, they may have persisted for a substantial period of time. While similar properties can influence stability in the number system, dissimilar ones can motivate both conservation and change. Consider languages whose names for numbers are relatively long expressions describing finger-counting, like those of the Yuki mentioned in Chapter 5. In Dano, a language spoken in the Eastern Highlands of Papua New Guinea, the term for nine is ade hela osu’ livo ade hela seta’ve seta’ve ivo, which means something along the lines of our hand on one side being finished, our hand on the other side having four, while the term for sixteen is ade hela osu’ livo ade hela seta’ve seta’ve ivo gizede hela osu’ livo gizede hela hamo’ ivo, or both our hands and our foot on one side being finished, our foot on the other side having one.12 The length of expression increases if what is being said is not a single number but a counting sequence. When number words are bulky, speakers Saxe, Cultural Development of Mathematical Ideas. Saxe, p. 86. 12 Strange, ‘Dano, Papua New Guinea’. 10 11

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may prefer to express numbers by means of gesture. Because expressing numbers with gesture precedes their expression in language, this may reflect the persistence of a cultural habit. However, gesture is also quicker, less ambiguous, and more useful in counting. As gesture conserves, the linguistic form would be under pressure to become more like it. And indeed, bulky number words become shorter, not only through the diachronic change described in Chapter 6 but by omitting parts understood through context.13 Contrast and congruence are neither advantageous nor disadvantageous in themselves, merely circumstances informing numerical outcomes. Compare numbers in Chinese and English (Table 11.2). Both are perfectly regular in their decimal structure, and this is structure is clearly expressed by both notational forms. Chinese spoken numbers are identical to their written counterparts, congruence that makes their decimal structure clear, accessible, and easily learned. English spoken numbers, however, are not identical to their written counterparts. The numbers one to ten are unanalyzable syllables, the numbers eleven to nineteen are irregular compounds, and the numbers twenty and higher are fairly regular compounds. This mismatch makes the decimal structure a bit less clear and correspondingly less accessible and intuitive. Congruent reinforcement in Chinese written and spoken forms of number but not in English is thought to explain, at least partially, why Chinese speakers perform mathematical tasks slightly better than English speakers do.14 ୍ ஧ ୕ ᅄ ஬ භ ୐ ඵ ஑ ༑

yi èr san sì wu liù qi ba jiu shí

༑୍ ༑஧ ༑୕ ༑ᅄ ༑஬ ༑භ ༑୐ ༑ඵ ༑஑

Chinese shí yi shí èr shí san shí sì shí wu shí liù shí qui shí ba shí jiu

୍ⓒ ஧༑ ୕༑ ᅄ༑ ஬༑ භ༑ ୐༑ ඵ༑ ஑༑

yi bai èr shí san shí sì shi wu shí liù shí qui shí ba shí jiu shí

1 2 3 4 5 6 7 8 9 10

one two three four five six seven eight nine ten

11 12 13 14 15 16 17 18 19

English eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen

100 20 30 40 50 60 70 80 90

hundred twenty thirty forty fifty sixty seventy eighty ninety

Table 11.2. Chinese and English numbers. The decimal structure is identical between notational and lexical numbers in Chinese (left) but not English (right). This is thought to make the former easier to acquire at an earlier age and easier to manipulate throughout life, suggesting the co-influence between the material and linguistic forms has cognitive implications. Data compiled from multiple sources.

As the Spanish encounter with Easter Island suggested, differences between number systems can provide useful contrasts. Writing to Honolulu’s Ka Nonanona newspaper in 1842, a correspondent noted that mastery of Western numbers was critical if the Hawaiian people were to negotiate economic concerns with entities from outside the 13 14

Crockett, ‘Moi (Dao), Papua, Indonesia’. Cantlon and Brannon, ‘Adding up the Effects of Cultural Experience on the Brain’.

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Kingdom and operate successfully within their legal and political frameworks. 15 Similar considerations of parity and enfranchisement likely motivated the Akkadians and Elamites to adopt Sumerian written numbers, though both would also adapt them— the Akkadians to a lesser degree, the Elamites to a greater one—to reflect the decimal structure of their indigenous numbers. The contrast between sexagesimal and decimal organization was also possibly a factor in noticing and explicating place value, whose emergence in the Old Akkadian period (2350–2250 BCE)16 coincided with the adaptation of cuneiform writing to the Akkadian language (Fig. 10.8). That is, while place value clearly drew upon the linearity and magnitude ordering of previous forms, reformulating that existing structure as place value may have required insights acquired from the contrast of organizing bases. Contrast is not useful merely because it illuminates exploitable properties. Consider the materials sched, a resource for innovation17: In design companies a materials shed, sometimes called an ideas cart, is often relied on for stimulation during the ideation phase of design. A materials shed is a collection of artifacts like mechanisms, fabrics, patterns, iconic designs and toys or gadgets that a designer, in the early part of research, is encouraged to visit. Viewing the thousands of items filling the shed, someone in search of inspiration might respond to an odd metallic texture, an unusual gear configuration, a suction device or a mechanism exploiting a physical principle in an uncommon way. The aspect that triggers interest need not be an element that would ever work in the final design or a part of something central to the design. Its role is to provoke an idea that had not been considered before, an idea that helps the designer realign how he or she thinks or that arouses consideration of candidates outside the norm.18

Random stimuli may function as distraction, allowing our creative and patterncompleting mechanisms to unfold. But they may also shake loose something we’ve seen in numbers: Part of numbers’ implicit component is habit, and habituation to the way a particular material form works both limits us from using it differently— functional fixedness—and influences the new forms we select and how we use them. But habit, custom, expectation, and knowledge also tend to blind us to possibilities for change. Cognitive scientist David Kirsh expresses it even more strongly: When it comes to creativity, ‘[f]amiliarity is the enemy’ because ‘the reverse effect often happens. Past ideas, or idea analogs, become fixated. The reviewed solutions bias thinkers to look for variants on existing solutions or to use existing search spaces, rather than invent new ones’.19 Like the materials shed, contrastive properties in numbers have destabilizing and broadening effects, making the incorporation of new material forms Lagaro, ‘Ka Waiwai o ka Helu [The Value of the Number]’. Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin; Robson, ‘Mesopotamian Mathematics’. 17 Kirsh, ‘The Importance of Change and Interactivity in Creativity’. 18 Kirsh, p. 22. 19 Kirsh, p. 22. 15 16

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not just the source of numerical properties but a primary mechanism for their elaboration. And here at last, we might also find a satisfactory raison d’être for the correlation between material culture and numerical elaboration, an answer to the mystery of what it is about material culture generally that stimulates numerical creativity, since it has not been found in any of the places we’ve looked, like resourcing and sustainment strategies.20 It is not just that material complexity demands we manage it somehow, or that materiality and numbers become habitual or preferential problem-solving resources and strategies, or even that devices let us count to the higher numbers that potentialize numerical relations, though all of these are true. Rather, it is perhaps that material culture generally acts like a materials shed, one that as it expands and elaborates becomes more random and less connected to the cognitive domain of numbers. And unlike the designer who visits occasionally in search of inspiration, we inhabit our materials shed. However inured to its many attractions and distractions we become, it is nonetheless our environment, our cognitive landscape, poised always to challenge our status quo and stimulate our creativity, whenever we are ready.

OTHER MATERIALLY INFLUENCED CHANGE Useful contrast would not have been the only mechanism of change in the cognitive system for numbers in the Ancient Near East, as interacting with materiality would also have changed psychological processing and behaviors: x

Interacting with discrete, linear material forms would have increased acuity and linearity in the mental component of numbers,21 facilitating the conceptualization of higher quantities in regularized and productive ways.

x

Repeating behaviors like finger-counting,22 manipulating tokens, and writing would have influenced their becoming habitual and automatic. This in turn would have freed resources of attention and working memory and allowed them to be repurposed to the pattern recognition and categorization that factor into forming new concepts.

20 Divale, ‘Climatic Instability, Food Storage, and the Development of Numerical Counting: A Cross-Cultural Study’; Epps et al., ‘On Numeral Complexity in Hunter–gatherer Languages’. 21 Fischer, ‘Finger Counting Habits Modulate Spatial-Numerical Associations’; Göbel, Walsh, and Rushworth, ‘The Mental Number Line and the Human Angular Gyrus’; Núñez, ‘No Innate Number Line in the Human Brain’; Piazza et al., ‘Education Enhances the Acuity of the Nonverbal Approximate Number System’. 22 Dehaene et al., ‘Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence’; Domahs et al., ‘Embodied Numerosity: Implicit Hand-Based Representations Influence Symbolic Number Processing across Cultures’; Zamarian, Ischebeck, and Delazer, ‘Neuroscience of Learning Arithmetic: Evidence from Brain Imaging Studies’.

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x

Declarative information like relations between numbers, made explicit, visualizable, and manipulable through technologies like tokens and notations, would have further involved the angular gyrus, particularly in its function in recalling arithmetic facts.23 Memorizing and recalling arithmetic facts would also have changed how numbers were understood and calculations performed.

x

Arithmetical operations would have recruited executive functions like shifting, the ability to switch flexibly between tasks or mental sets; updating, the ability to monitor and manipulate the contents of working memory; and inhibition, the ability to override prepotent or dominant responses in pursuit of a goal.24 These are important to performing arithmetical tasks, which is why there is a correlation between executive functioning and mathematical performance.25 Increased recruitment of the executive functions, in turn, would have increased goal-directed behavior, allowing people to do things like rearrange tokens into new combinations, perform more complex calculations, and seek new relations and patterns in numerical data.

x

Highest number counted would have increased as material forms were recruited, a straightforward matter of using devices with the capacity for counting to higher numbers.26 An increased highest number counted, in turn, would have made it more likely the relations between numbers would have been discovered, explicated,27 and codified as artifacts, behaviors, algorithms, and language.

x

Numbers would have acquired properties like discreteness, linearity, and grouping. As they became manipulable and gained relations, they would have changed from quantity equivalences to collections related to the objects they enumerated, then to objects related to each other as well as the objects they enumerated, and finally to entities in a relational system characterized and defined by their relations to one another.

23 Dehaene et al., ‘Three Parietal Circuits for Number Processing’; Grabner et al., ‘Individual Differences in Mathematical Competence Predict Parietal Brain Activation during Mental Calculation’; Grabner et al., ‘To Retrieve or to Calculate? Left Angular Gyrus Mediates the Retrieval of Arithmetic Facts during Problem Solving’; Grabner et al., ‘Fact Learning in Complex Arithmetic and Figural-Spatial Tasks: The Role of the Angular Gyrus and Its Relation to Mathematical Competence’; Ramachandran, A Brief Tour of Human Consciousness: From Impostor Poodles to Purple Numbers. 24 Miyake et al., ‘The Unity and Diversity of Executive Functions and Their Contributions to Complex “Frontal Lobe” Tasks: A Latent Variable Analysis’. 25 Bull and Lee, ‘Executive Functioning and Mathematics Achievement’; Bull and Scerif, ‘Executive Functioning as a Predictor of Children’s Mathematics Ability: Inhibition, Switching, and Working Memory’; Van der Ven et al., ‘The Development of Executive Functions and Early Mathematics: A Dynamic Relationship’. 26 Overmann, ‘Material Scaffolds in Numbers and Time’. 27 Beller and Bender, ‘Explicating Numerical Information: When and How Fingers Support (or Hinder) Number Comprehension and Handling’.

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As numbers elaborated, their utility would have improved, increasing the likelihood of their being used and integrated into social purposes. Utility and use would have positioned numbers as a cognitive technology, a body of knowledge structuring comprehension and behavior.28 The more familiar and habitual a cognitive technology becomes, the more likely it will be extended to new domains and integrated in specific behaviors and cultural practices.29 And the more ingrained in culture numbers become, the more likely they are to be elaborated; their usefulness may intensify the need to develop and extend them.

An outline of systemic change is provided as Fig. 11.2. The Piagetian notion of a sharp distinction between abstractness and concreteness in societal modes of thinking has been replaced with the idea of numbers becoming elaborated through the incorporation of new material forms. Elaboration is understood as change in numerical properties related to the material forms used to represent and manipulate them, the amount of information contained by the system, and the relative proportion of explicit representation supplied by the material form and implicit knowledge and neurological reorganization supplied by the user. The timeline differs as well: Where Damerow suggested one-to-one correspondence emerged at 10,000 BCE and there were no numbers prior to that point,30 I place one-to-one correspondence in the Late Upper Paleolithic (Chapter 8) and note that the relations between tokens exclude their use in one-to-one correspondence, certainly by the mid-to-late 4th millennium BCE, and possibly earlier (Chapter 9). Written notations, along with the ability to record the steps of calculations as descriptive narratives made possible by writing and the literacy that emerged around 2000 BCE (Chapter 10), expanded numerical elaboration even further to produce a complex mathematics by the Old Babylonian period (1900–1600 BCE). Gaining properties from multiple material forms means no single form has all the properties numbers eventually possess. This not only distributes numbers over multiple forms, it makes them independent of any particular form. I am not suggesting this independence ever means material forms are no longer required, as this would only make numbers conceptual units without the physical reality that makes them meaningful.31 In fact, I see numbers as remaining materially bound no matter how elaborated they become, the converse of the idea they are abstract from their inception. Numbers depend on material forms to give them tangibility and visualizability, the properties that make them accessible, comprehensible, and manipulable. In turn, material forms structure and organize numbers in a way that informs both how we acquire them and what we understand them to be.32 28 De Cruz, ‘An Extended Mind Perspective on Natural Number Representation’; ‘Are Numbers Special? Cognitive Technologies, Material Culture and Deliberate Practice’. 29 Hodder, Entangled: An Archaeology of the Relationships between Humans and Things. 30 Damerow, ‘Number as a Second-Order Concept’, 139–148. 31 Renfrew, ‘Commodification and Institution in Group-Oriented and Individualizing Societies’, p. 98. 32 Schlimm, ‘Numbers through Numerals: The Constitutive Role of External Representations’.

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Fig. 11.2. The elaboration of numbers. As new devices were added to the system, they would have influenced numerical organization, capacity, and the amount and types of relations and operations possible. Solid lines represent unambiguous evidence, dashed lines possible evidence. Created by the author using data from multiple sources.

Distribution over multiple forms and independence from any one form are interesting qualities. They make it possible for us to apprehend 75 as the same number, regardless of whether it is represented by fingers, tally notches, Mesopotamian tokens, or any of the notational systems shown earlier. They enable us to use multiple forms for our numbers, albeit with the asymmetric perspective associated with enculturation into number systems with relatively more or less elaboration, mentioned earlier for Spanish and Easter Island numbers in the late 18th century CE. Distribution and independence mean that when the ancient Greeks used at dot matrices to visualize and investigate numerical properties, their concept of numbers as relational entities allowed them to focus on constitutive elements like the literal square of 2 × 2 dots comprising the number four.33 Neolithic Mesopotamians arranging tokens into a two-by-two square (Fig. 11.3) were more likely to have conceived them as a collection, rather than a coherent entity with both external and internal relations, despite the similarity of form. Distribution and independence also mean that when we look at the material forms in numerical systems belonging to other cultures or ancient societies, we tend to superimpose our number four onto their material representations, whose representational intelligibility obscures substantial differences in the associated concepts.

33

Klein, Greek Mathematical Thought and the Origin of Algebra.

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Fig. 11.3. Similar material forms, different numerical concepts. (Left) Ancient Greek mathematicians used dot matrices to visualize the internal relations of numerical entities, like the fact that four is composed of two times two. The symmetry of such matrices led to such arrangements being called squares, the term still used today to characterize the ideas of multiplying a number by itself (square) or finding its inverse (square root). (Right) Neolithic Mesopotamians could have placed spherical tokens into a similar arrangement but were more likely to have understood them as a collection comprised of four elements. Tokens adapted from Englund, ‘An Examination of the “Textual” Witnesses to the Late Uruk World Systems’, p. 29, Fig. 24.

Their independence also means that numbers are ultimately irreducible to all the material forms used to represent and manipulate them, even if those devices are, collectively, the source of their various properties. Numbers are also partly behavioral, bound to what we use them for, as well as how we use them. They are expressed not only materially and behaviorally but linguistically as well, and depend on knowledge and habits that are socially imparted. They require ontogenetic maturity and conceptual knowledge, and sometimes skills involving neurological reorganization.

DISTRIBUTION, INDEPENDENCE, AND OTHER SO-CALLED ABSTRACT QUALITIES

Within mathematics, the abstract–concrete distinction has a longstanding and relatively straightforward definition. In Higher Arithmetic, one of several textbooks by 19th-century mathematician James B. Thomson and one of the first mathematical texts translated into the Hawaiian language, numbers are concrete if they are ‘applied to particular objects, as peaches, pounds, yards, &c.’, and they are abstract when they are not, as in the expression ‘two and three are five’.34 Past discussions of how numbers changed from concrete to abstract have focused on separating the representation of quantity from that of commodity, conjoined in tokens and separated in writing.35 The technological change meant numbers were freed of a complication that made their further elaboration more challenging, assuming it were even feasible. The change also reflects what appears to be a relatively common strategy for conceptual elaboration: Material forms are a substrate for conceptual explication and elaboration. That is, material forms an34 Thomson, Practical Arithmetic, Uniting the Inductive with the Synthetic Mode of Instruction: Also, Illustrating the Principles of Cancelation, p. 261. 35 Malafouris, ‘Grasping the Concept of Number’; How Things Shape the Mind: A Theory of Material Engagement.

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chor and stabilize our concepts,36 accumulate and distribute our cognitive effort across space and time,37 and enable us to appreciate and visualize patterns. When we divide, rearrange, and recombine material forms, the concepts associated with them have an opportunity to change as well. Certainly, once freed of the encumbrance of commodity, numerical notations would have represented numerical information more clearly, something that would have facilitated the elaboration of numbers as a mathematical system. Yet abstract is one of those fuzzy terms that means different things in different contexts. Beyond meaning a number that refers to other numbers rather than any particular object and as I mentioned in Chapter 4, what abstract means differs according to its context of use: psychological processing, mathematical realism, change in written form,38 distillation to an essence, or the goals and uses of knowledge.39 And I have offered yet another way to use the term, the distribution of numbers over multiple forms used for representation and manipulation and their seeming independence of any particular form. Numbers might be abstract under several of these definitions, but we need to be clear about which meaning(s) we intend, and whether any meaning entails a state of previous concreteness. For the sense of abstract wherein a number refers to other numbers rather than any particular object, the idea has been that Mesopotamian numbers became abstract once they took the form of written notations sometime in the mid-to-late 4th millennium BCE. This is problematic for several reasons. One is the ability of numbers to refer to each other was already fairly well developed in the tokens, not to mention implicit to the ordinal relations of tallies and fingers; thus, it cannot accurately be said to have happened when and solely because writing was invented, though admittedly, writing numbers enabled a greater elaboration of their relations than was possible with previous technologies. Another reason is that written notations did not act alone in this capacity. Certainly, handwriting effects facilitated the conceptualization of numbers as entities, and notational concision allowed relational information to be collected in tables and learned through the practice of rewriting tables as part of scribal training. However, given that numbers do not come preloaded with relations, the relations were plausibly worked out with tokens, since a manipulable form is required to make relations visible and explicit, and notations are not manipulable. Writing has also been thought to have made numbers abstract by representing quantity and commodity separately, as if extracting numbers from commodity and non-numerical writing from quantity refined each of the impurity represented by the other. If the complicating factor was conjoined representation, it must be pointed out that neither fingers nor tallies did this, since they did not represent commodity at all; instead, the elaboration of relations between numbers would have been inhibited by properties of these material forms, like their relative lack of manipulability. Quantity Hutchins, ‘Material Anchors for Conceptual Blends’. Hutchins, Cognition in the Wild. 38 Schmandt-Besserat, ‘The Earliest Precursor of Writing’, p. 50. 39 Høyrup, In Measure, Number, and Weight: Studies in Mathematics and Culture. 36 37

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and commodity representation were conjoined in tokens, early impressions, and protocuneiform notations. Seals could accompany these technologies, carrying information that in some cases might have also implied the type of commodity being enumerated. Handwritten signs for non-numerical language appeared shortly after the first impressions and would become increasingly important to representing the objects being enumerated. Undoubtedly, this representational separation—a material abstraction—let non-numerical writing and numerical notations be independently elaborated as literate and mathematical traditions. However, it’s unclear the Babylonians ever understood numbers as not referring to particular objects or matters like area and time. Even toward the end of the cuneiform period, which occurred several millennia after the representational separation, the commodity that numbers referred to was either specified by non-numerical signs or understood through context. The cuneiform numbers and numerical tables perhaps came the closest to being numbers that referred only to other numbers, the former as a system used to transform quantities between the metrological systems for specific objects or types of objects, the latter as compilations of relational data. Ancient Greek mathematicians, for all their theoretical investigations of numbers and numerical properties, thought numbers referred to things. Aristotle, for example, so thoroughly identified the everyday numbers used in counting and calculating, arithmos, with the objects they counted that the ten counting horses differed from the ten counting dogs, even though both had the same quantity.40 Greek philosophers were concerned with the ontological status of number in a way that Babylonian mathematicians had not been.41 But this yielded a foundational role for numbers in Greek ontology, where they connected the sensible realm of objects to the nonsensible domain of ideals.42 This indelibly associated the two: Numbers didn’t just count objects but were part of the machinery causing things to exist. Such strong associations between numbers and the things they counted are not really surprising, given the wealth of mathematical ideas the Greeks inherited from the Babylonians and the amount of time it can take for conceptual elaboration, especially ideas breaking radically from past thought. In fact, it was not until the end of the 19th century CE that numbers were conceived in a way avoiding their referring to objects at all. This development required axioms capable of defining numbers as a natural set,43 something involving several additional millennia of post-Greek thought. A final note about abstractness and writing concerns the loss of depictiveness Schmandt-Besserat asserted as a form of becoming abstract. This quality did characterize non-numerical writing, a change apparent in comparisons of later written forms and the pictograms and ideograms preceding them. However, such change cannot be said to have characterized Mesopotamian numerical notations, as these continued to be Klein, Greek Mathematical Thought and the Origin of Algebra. Høyrup, In Measure, Number, and Weight. 42 Klein. 43 Crossley, The Emergence of Number, p. 44; Peano, Arithmetices Principia: Nova Methodo Exposita, p. 1. 40 41

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instantiated by countable elements throughout the lifespan of the cuneiform script. Some numerical notations—those requiring the instantiating elements be counted— would eventually lose their depictiveness, if this term can apply to instantiation as a representational mode with the same meaningfulness it has for resemblance. Numerical notations would ultimately become ciphered forms that did not require counting elements. However, this change in written numerical form would not take place until long after cuneiform had ceased being used. I am accordingly reluctant to characterize numerical notations as abstract, tokens as concrete, and the representational separation afforded by writing as the occasion and mechanism for distinguishing the two. I find the use of the term abstract often uncomfortably imprecise, and the ideas of a previous concreteness and a sharp distinction between the two untenable. For numbers, writing is also only a part of a much longer technological sequence. Separating quantity and commodity as notations and pictures doesn’t recognize contributions like handwriting effects and notational concision. These would have helped numbers become conceptualized as entities and redefined through their relations to one another, things that strike me as more than simple nuance to the idea of an abstract–concrete distinction occasioned by writing. Further, as an abstraction, the representational separation was not just mental, but involved material, behavioral, and psychological factors. For example, all the material forms in the longer sequence of devices, even fingers, could be used away from the objects they enumerated, helping influence the conceptualization of numbers as something other than object properties. Critically, focusing on writing and the representational separation it afforded misses the insights the longer material sequence might provide: what numbers are as concepts; how numbers become what they are as concepts; how material devices help elaborate numerical intuitions into explicit concepts; and what materiality does to mediate our access to, and conceptualization of, numerical concepts. These things are not easily seen when numerical notations and the representational separation are considered in isolation. I want to mention some of the other meanings of the term abstract and whether they entail a former state of concreteness. Numbers are abstract in the psychological sense, as they are realized and changed through the cognitive process of abstraction. Since they begin as judgments that sets of objects share quantity, an abstraction, a previous state of concreteness is excluded. While the Platonic sense in which numbers are purported to be abstract is unfalsifiable, it too excludes a previous state of concreteness, though for different reasons. Granted, the practical concerns of the Babylonians would have made their mathematics more concrete than the science of the later Greeks, as the theoretical concerns of the Greeks would have made their mathematics more abstract. This use of the term abstract has the sense of being concerned with what numbers are as concepts, including their relations to one another, a matter that seems to have occupied the Greeks much more than their Babylonian predecessors. But it’s also worth noting that the Greeks built upon what the Babylonians realized, and both used materiality in ways that influenced the content, structure, and organization of their number concepts. Another way to frame this is to see both mathematical traditions as different points on the same process of numerical elaboration, a spectrum we inhabit as well with our current conceptualization of number. On this view, the Babylonians

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and Greeks had more in common than not, and we to both of them as well, making it difficult to endorse labeling any one concrete or any other abstract. Let’s examine more closely the idea of numbers being distributed over multiple material forms in a way making them independent of any particular form. Even with a single material form, even a highly concise one like writing, a number is potentially distributed over multiple forms. In the number 75, for example, there are two signs, 7 and 5. The 7 has the ciphered meaning seven but is understood as seventy because it appears in the tens place, and the 5 has the ciphered meaning five and only five because it appears in the units place. This is about as concise as such representations can get, apart from the infeasible hypothetical system in which all conceivable numbers are represented by unique signs. But despite its being distributed over two signs, 75 is generally conceived a coherent number. I mention this because one of the reasons the Mesopotamian numbers associated with tokens have been thought of as concrete is their similar distribution between multiple counting systems. In Aristotelian terms, the ten used to count people in Sexagesimal System S would have differed from the ten used to count grain in ŠE System Š in whatever part of its essence did not relate to the quantity ten. But what has been assumed in the Assyriological literature exceeds this: The Mesopotamians who used tokens for accounting are presumed incapable of recognizing even the quantity ten as being the same, echoing Bertrand Russell’s sentiment that it must have taken ‘many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2’.44 Nor was this the limit of their supposed numerical incompetence. Polyvalence and context-dependent meaning have also been used to support the idea the Neolithic Mesopotamians lacked abstract numbers. Polyvalent signs had a different numerical value according to their context of use, like N14 meant ten N01 in the system for counting most discrete objects but six N01 in the system for counting grain. The numerical value of N14 thus depended on its context. In other cases, the same numerical value could be represented by different signs, as 75 would have been represented by one N34, one N14, and five N01 in the system for counting most discrete objects but one N45, two N14, and three N01 in the system for counting grain. Against the idea multiple systems, polyvalence, and context dependence represented concrete numbers while notations marked abstract concepts, it must be noted that multiple counting systems continued in use long after notations were invented, since place value emerged even later to integrate the different counting systems. Further, as a basis for the presumed absence of abstract numbers, polyvalency and context-dependence quickly lead to absurdity: ‘If … polyvalence and context-dependence imply an absence of abstract number concepts, then paradoxically, the quasi-literate Uruk accountants would be less numerate than the average Sumerian who did not use texts, only number words’.45 Regarding the use of tokens and numerals in the mid-to-late 4th millennium BCE, surely ‘the accountants and scribes who used them were able to manage complex administraRussell, Introduction to Mathematical Philosophy, p. 3. Chrisomalis, ‘Evaluating Ancient Numeracy: Social versus Developmental Perspectives on Ancient Mesopotamian Numeration’, p. 4. 44 45

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tive tasks, and it is implausible that they did not recognize that “8 sheep” and “8 bushels of grain” had something in common’, their quantity.46 I propose the Mesopotamians could and did understand numbers despite the distribution of these concepts over multiple forms. Beyond avoiding the ridiculous ends the converse can reach, I base this conclusion on the circumstance the Mesopotamians retained and used multiple material forms for their numbers, just as we ourselves do. I assume Mesopotamian people would have counted on their fingers throughout their involvement with numbers, since this behavior has been observed across the full spectrum of numerical elaboration just about everywhere. The behavior is so common, in fact, the instances where it has not been specifically recorded plausibly constitute something seen but thought unremarkable. Tallies continued in use, as attested by a 2nd-millennium BCE poem, 1st-millennium BCE artifacts, and, if the Levantine devices were in fact tallies, by Upper Paleolithic and Epipaleolithic artifacts as well. And tokens persisted long after notations were invented, at first because there was no alternative for calculating, and later, once the relations and algorithms needed for calculating with notations were invented, because they were easier, like we might resort to a calculator. The side-by-side use of multiple technologies suggests the Mesopotamian concept of number was distributed over them too, just like ours is over the multiple devices we use. And this, in turn, implies the use of multiple counting systems with polyvalent, context-dependent signs would not have represented concepts of numbers that were particularly concrete in their nature or poorly understood by their users. What about the relative necessity of involving material forms as a quality characterizable as abstractness: abstract if they don’t need material forms, concrete if they do? Most of us do, in fact, need material representations to manipulate numbers, something experientially verifiable through the relative ease of extracting square roots to five decimal places with written algorithms or a calculator, relative to not involving any material form. In Mesopotamia, even when numbers were written and had become abstract in referring primarily to each other, they depended on material representation to make them tangible and thus manipulable. Numbers remained just as concrete—in the sense of depending on materiality—after the representational separation as they had been before it. There is also a significant contiguity between written and non-written forms of numbers. Tokens and the earliest notations, for example, differed in their manipulability but were otherwise identical, apart from the trivial matter of the one being an impression of the other. This contiguity is not often remarked, partly because written forms have tended to be treated as ontologically different enough to require separate analysis from non-written forms, and partly because of a dominating belief that numbers are mental phenomena. As concepts, numbers have undoubtedly changed over time in their content, structure, and organization. This change occurred throughout the sequence of material devices and because that sequence expanded to include new forms. That is, numbers didn’t become abstract so much as they gained properties. As the properties became more explicit and elaborated and some were transferred between forms as behaviors, 46

Chrisomalis, ‘The Cognitive and Cultural Foundations of Numbers’, p. 502.

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habits, and expectations, the knowledge required to understand and use the forms increased. This in turn enabled the forms to become increasingly concise. As numbers were becoming distributed and independent and their concise representations required more user knowledge, they acquired properties that, in involving a larger requirement for user knowledge, could be considered to make them more abstract. Nonetheless, I don’t think this ever makes them intangible. Many people, if not most of us, depend on material forms for numerical representation and manipulation. We acquire and understand numbers through and because of the forms we use; what we understand numbers to be is significantly influenced by the forms we use. Thus, numbers remain materially bound, even at their most elaborated.

CHAPTER 12. CONCLUDING REMARKS AND QUESTIONS At a recent Cambridge conference on early writing systems, I was asked a thoughtprovoking question, the gist of which was this: Did writing represent a system containing a historically unprecedented amount of information? The time period specified for this, if I heard correctly and memory serves, was 3200 BCE. My answer, both then and now, is no. The information contained by the system needs to be considered separately from the development of its ability to contain information and the information an individual would need to access it. The information-generating and -storage capacity of precursor technologies like tokens would also have to be taken into account. By 2000 BCE, interaction with, and change in, the neurological, behavioral, and material domains of writing had produced a tool capable of expressing language with fidelity and interacting at a speed approaching that of thought. This expressive capacity would have profound implications for subjecting human thought to analysis and communicating it across space and time.1 In terms of the information contained by the system, writing was storing and making potentially available domains of history, literature, politics, religion, medicine, mathematics, astronomy—even personal correspondence— like never before. For numbers, written notations had given them attributes of entitivity and concision, enabling the creation of relational tables, facilitating the use of relations in computation, and transforming numbers into a complex relational system. Non-numerical writing was being used to narrate instructions for manipulating numbers, enabling operations to become explicit, codified, and more complex. No other invention has had anywhere near a comparable impact, even as things like computers and networks have lately made writing as a system go faster and further and become more capable of collecting, storing, recalling, distributing, analyzing, rearranging, transforming, and calculating information. At 3200 BCE, however, and for many centuries thereafter, the amount of information the system contained would not change remarkably, relative to what had preceded it. Writing would essentially contain what tokens had contained before it, although where tokens had conjointly encoded commodity and quantity, writing and numerical notations would separate the two. Nor would the information needed to participate in the system have been remarkable, as in its initial incarnation, writing would 1

Olson, The World on Paper: The Conceptual and Cognitive Implications of Writing and Reading.

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have been fairly accessible. Drawing lines on clay would not have required learning a lot of new information or acquiring new skills, nor recognizing characters a significant neurological reorganization, relative to the amount of these things needed, say, a thousand years later. The meaning of the small pictures comprising the set of signs in early writing was a function of their resemblance to familiar objects or conventions predating writing, some of which may have been used in the so-called complex tokens (Chapter 9).2 The information required to recognize signs would have been further mitigated by the lexical lists, sign compilations appearing almost as soon as the signs did themselves. While lexical lists have been described as ‘dictionaries’ and ‘teaching tools’,3 at the beginning of writing they most likely functioned more as codebooks, with signs listed and meanings memorized in a mutually consistent order enabling the meaning of unfamiliar signs to be obtained.4 Over the centuries that followed, change in the information system would be more a matter of reorganizing behaviorally, neurologically, and materially (Chapter 10) than accumulating information, either in the system itself or in terms of what its users had to know. For behaviors and brains, change involved existing capacities like fine motor control, object recognition, and language. Change in the material form of writing consisted of matters like the loss of depictiveness enabled by recognizing characters by their features and adding visual clues to specify pronunciation or word type. The system appears to have crossed a critical threshold around 2000 BCE, when two things happened. First, applications and experimentation with writing expanded so dramatically, the threshold has been called a ‘revolution’.5 Second, the kinds and rate of change characterizing the reorganization of the system suddenly slowed. For writing, this is where I would place the true beginning of its historically unprecedented ability to accumulate information, with change leading up to that point serving to develop and actualize the information system itself. For numbers, writing was only the latest twist in a much longer technological sequence. It would not greatly increase the amount of numerical information the system retained, since tokens were already serving this purpose. Being written would increase informational longevity to a historically unprecedented degree; some has even lasted till today, something only tokens contained unbroken in sealed bullae could ever hope to accomplish—assuming any exist, and we were willing to break them open to verify their contents, and those contents were to correspond to external markings that could attest to their numerical meaning. And the amount of information someone would need to know in order to use numerical impressions would not have changed, at least initially. Both numerical impressions and the proto-cuneiform notations that followed them were identical, or very nearly so, to their token predecessors. Tokens were threedimensional objects, impressions were the three-dimensional imprints of tokens in clay, Schmandt-Besserat, Before Writing: From Counting to Cuneiform. Veldhuis, History of the Cuneiform Lexical Tradition, pp. 16 and 56. 4 Wagensonner, ‘Early Lexical Lists Revisited: Structures and Classification as a Mnemonic Device’. 5 Veldhuis, ‘Cuneiform: Changes and Developments’, p. 3. 2 3

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and proto-cuneiform notations were three-dimensional imprints made with styli rather than tokens. These forms differed primarily in their manipulability: Tokens could be rearranged; impressions and proto-cuneiform notations, once made, were fixed. Beyond this difference, the knowledge required to use the different forms and the neurological basis for recognizing them would have been difficult to distinguish. As cuneiform characters, notations were concise beyond the level of any of the previous forms used for representing and manipulating numbers. This concision would have required users to reorganize neurologically and learn things not explicitly represented, like relations between numbers and algorithms for performing operations with notations (Table 12.1). These changes would do more than facilitate the recognition and use of cuneiform notations: they would ultimately enable the reconceptualization of numbers as entities in a relational system. But the increase in the amount of systemic information would still have been fairly limited, as some of the relations and many of the algorithms had been gained through the use of predecessors like tokens. And the increase in facts and figures needing to be memorized would have been mitigated by the tables being available for purposes of storage and recall. Nr 1 2 3 4 5 6 7 8 9 10

Sumerian diš min eš limmu ya aš imin ussu ilimmu u

ED IIIb X X X X X

Old Akkadian X X X X X

X

Table 12.1. Cuneiform numbers. As listed in The Pennsylvania Sumerian Dictionary, six of the cuneiform numbers one through ten are attested by the Early Dynastic IIIb period (2500– 2340 BCE). All but three are attested by the Old Akkadian period (2340–2200 BCE). The Pennsylvania Sumerian Dictionary does not list period information for five, eight, or nine. The data were drawn 29 March 2019.

Even as I quibble over when, what, and how much, I accept the larger point behind the question: Writing, numerical and otherwise, transformed our relationship with information. Knowledge would no longer be encoded and intergenerationally transmitted only through mechanisms like tools, behaviors, speech, and organic memory, things that on balance are relatively imprecise, perishable, and subject to reproduction error when compared to the specificity, fidelity, and persistence of writing. And for numbers, written form meant they would no longer be conceived as collections. The idea written notations would ultimately transform numbers raises five new questions, answered in this final chapter: whether the process whereby Mesopotamian numbers were realized and elaborated can be said to be a universal one; whether numbers originated once or many times; whether and the degree to which written notations and precursor technologies like tokens are essential to mathematical elaboration; whether the increase in implicit knowledge required to use written notations makes numbers a men-

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tal phenomenon; and why, if material forms are so essential to what numbers are as concepts, we often don’t notice what the material forms do, either when using them ourselves or as a matter of historical emergence.

SOME ANSWERS TO THE QUESTIONS POSED Can the way Mesopotamian numbers were realized and elaborated be said to represent a universal process? The process does appear universal, if only at its topmost level of detail. All known systems of numerical notations are similar in mixing concise, explicit representation with an implicit component of knowledge required to understand them.6 Implicit structure and organization are so similar across number systems,7 the numerical meaning of unfamiliar notational forms can be recognized and interpreted even when the rest of the script or language is unknown or untranslatable. Recognition and interpretation are a matter of analyzing notations for the regularities indicating their relations, which in turn makes plain their conventions and unmasks their implicit component. The commonalities in the structure and organization of numerical notations generally suggest their properties emerge from using fingers and one- and two-dimensional devices to represent and manipulate numbers, with material properties persisting as behaviors, habits, expectations, and knowledge like they did in Mesopotamian numbers. This conclusion is further warranted by the commonality of human psychological processes, physiological attributes of human bodies, and the human behaviors with materiality underlying and informing the numbers, as well as the historically and linguistically attested use of fingers and other material forms for counting. Simply, all notational forms imply a prehistory of material devices whose sequence was the mechanism for accumulating and elaborating their properties. This is not to deny Mesopotamian numbers their unique aspects. Tokens and early notations conjoined the representation of quantity and commodity, a complication not found in other numerical traditions. Mesopotamian numbers were organized sexagesimally, where Maya numbers were vigesimal and Egyptian and Chinese numbers decimal. These systems differ in the details of the information explicitly represented by the notations and the information and neurological reorganization users must supply to understand and use them (Table 12.2). Chrisomalis’ typology was discussed in Chapter 10. Zhang and Norman’s cognitive taxonomy analyzes dimensionality according to the different physical properties used to represent it: quantity, shape, and position.8 In this taxonomy, cuneiform and Maya numbers are both (Quantity × Shape) × Position systems, with the parentheses around Quantity and Shape marking the conjoint use of repetition and form to encode numerical meaning. Egyptian is a Quantity × Shape system, Chinese a Shape × Shape. Though it is relatively limited, the variability that does exist in the structure and organization of notational forms and how they mix explicit and implicit components may ultimately be traceable to differences in the material

Zhang and Norman, ‘A Representational Analysis of Numeration Systems’. Chrisomalis, Numerical Notation: A Comparative History. 8 Zhang and Norman, p. 278. 6 7

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forms preceding notations, but investigating whether this is the case will have to wait for future study. System Tokens; protocuneiform

Within

Between

cumulative

additive

Egyptian

cumulative

additive

Roman Chinese Cuneiform Maya Hindu–Arabic

cumulative multiplicative cumulative cumulative ciphered

additive additive positional positional positional

‘75’

LXXV ୐༑஬ ਸ਼ೈൠ

75

Base



Ciphering

Implied Operation

60 Yes (౶ = 10)

(7 × 10) + (5 × 1)

10 Yes ( = 10)

(7 × 10) + (5 × 1)

10 10 60 20 10

(1 × 50) + (2 × 10) + (1 × 5) (7 × 10) + (5) (1 × 60) + (1 × 10) + (5 × 1) (3 × 20) + (3 × 5) (7 × 10) + (1 × 5)

Yes (L = 50) Yes (୐= 7) Yes (ೈ = 10) Yes ( = 5) Yes (7 = seven)

Table 12.2. Comparison of numerical notations. All notational systems mix explicit representation with an implicit component the user must supply, but all of them differ in the details. The columns labeled Within and Between are the intra- and inter-exponential dimensions defined in Chrisomalis’ typology.9 The Ciphering column does not list all of the ciphered forms used in a particular system: For example, in Roman numerals, in addition to L meaning 50, ciphering encodes X as 10 and V as 5, while in Chinese numerals, ୐ not only means 7, but ༑ means 10 and ஬ means 5. Chinese numbers are multiplicative– additive in Chrisomalis’ system, reflecting the use of explicit signs to characterize positional weight; these multiply the value they modify, and the results are added to achieve the total value. In Zhang and Norman’s system, the Shape × Shape designation for Chinese notes the use of explicit signs for numerical and exponential value without expressing the operations needed to derive total value, something the user must know. Neither schema characterizes the fact that Chinese written numerals are cumulative in the range of 1 to 3 and ciphered thereafter.

Did numbers originate once or many times? In the single-origin hypothesis, numbers emerged once and only once, spreading eventually throughout the world, and a single originating event accounts for the commonality and limited variability between systems.10 Much of the literature advocating a single origin dates to the 19th century, and few if any scholars seem to take it seriously these days. Nonetheless, it’s worth mentioning not just for its historical interest but because some of its more recent incarnations have proposed numbers to have spread from Mesopotamian, generally when a sexagesimal number system is found in a place like New Guinea.11 In the many-origins hypothesis, numbers have emerged lots of times in lots of places. This scenario assumes all societies to be equally capable of such invention. Under the many-origins hypothesis, numerical elaboration can be related to the elaboration of material culture generally, and each system can be viewed as having an individual potential for elaborating—or not— as governed by factors like demographics and situational context. The commonality Chrisomalis, Numerical Notation, p. 13. Seidenberg, The Diffusion of Counting Practices; ‘The Ritual Origin of Counting’. 11 Price and Pospisil, ‘A Survival of Babylonian Arithmetic in New Guinea?’. 9

10

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THE MATERIAL ORIGIN OF NUMBERS

between systems can be explained as a function of shared psychological, physiological, and behavioral capacities, and the limited variability as arising from different choices in, and combinations of, material forms. Many origins also fit the way numbers emerged in Mesopotamia, where between three and five independently originated number systems were coming in contact with each other at the beginning of the Neolithic (Chapter 7). With a single origin, it’s difficult to explain why the Sumerian, Akkadian, and Elamite systems differed, since these ancient systems would presumably be closer in time to the originating event and should be more similar to one another as a result. It’s also difficult to relate Mesopotamian numerical developments to the emergence of numbers elsewhere. People were inhabiting Australia between 50,000 and 65,000 years ago and South America by 20,000 years ago.12 Complex societies were forming in Mesoamerica about the time Mesopotamian mathematical elaboration was reaching its height, and they were flourishing by the time cuneiform culture was dying out.13 These developments occurred between the 3rd millennium BCE and 1st millennium CE, too short a period for any contact between Mesoamerica and Mesopotamia, given the distances and transportation realities involved. Given this timeline, if a single origin were true, numbers would have to be quite ancient—and in Australia, significantly older than even my provocative assessment would make them. Explaining why some number systems have very few numbers is another challenge: If the originating system counted to high numbers, contemporary restricted systems would represent the loss of capability; conversely, if the originating system counted within the restricted range, contemporary restricted systems would mean sustaining that state over a substantial stretch of time. Limiting a single-originating event to restricted numbers has other problems, like explaining why number systems in Eurasia and Africa are generally more elaborated than those in Australia and the Americas. I assume this well-established phenomenon has something to do with the movement of people around the planet and the length of time they’ve inhabited any particular place, which inform matters like sedentism and the new resourcing strategies it requires, as well as increases in population size, inter-group contact, and the associated demands for managing internal and external complexity. A final problem with positing a single origin for numbers is it assumes peoples around the world to be incapable of inventing numbers on their own, a suggestion frankly as unpalatable as it is untenable.14 There is positive evidence to support the idea of many origins, as for example, the use of different material exemplars to represent subitizable numbers across number systems. If the single-origin hypothesis were true, there should be less variability in this regard. But in fact, the smallest and presumably earliest numbers can show remarkable 12 Clarkson et al., ‘Human Occupation of Northern Australia by 65,000 Years Ago’; Davidson, ‘Peopling the Last New Worlds: The First Colonisation of Sahul and the Americas’. 13 Sharer and Traxler, The Ancient Maya. 14 Price and Pospisil, ‘A Survival of Babylonian Arithmetic in New Guinea?’; criticized in Bowers, ‘Kapauku Numeration: Reckoning, Racism, Scholarship, and Melanesian Counting Systems’.

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variability, even for near neighbors, as the number systems in Indonesia and Papua New Guinea do.15 This is perhaps unsurprising for a region with ‘the highest number of distinct languages in the smallest land mass in the world’.16 With many origins, there should be greater variability in the earliest phase of counting, less variability as number systems spread and come into contact with each other and converge, and both of these effects are well documented, with the interaction between the number systems of Mesopotamia being an instance of the latter. Language is another source of evidence suggesting numbers have emerged in many places and continue to do so today. Restricted numbers may be ‘one of the few areas in linguistics where present-day languages provide direct insight into the evolution of language’.17 The distribution of numerical features of language also conforms generally to the pattern of ancient human migration around the planet, suggesting the number systems in Australia and the Americas emerged more recently than those in Africa and Eurasia (Chapter 6). Problematically, however, the many-origins hypothesis glosses over relational factors like descent and diffusion; the first may have deep temporal roots, connecting Akkadian and Egyptian numbers through proto-Semitic, while the second can transfer numerical ideas, words, and technologies like the abacus, counting boards, games, and notations over impressive distance and time. Thus, few existing number systems are instances of truly isolated development. Between the two positions lies the several-origins hypothesis, the idea numbers originated a few times in a few places and spread everywhere else. In fact, this is what is thought to have happened for writing, which may have originated independently in as many as four places: Mesopotamia, Egypt, China, and Mesoamerica. However, for the majority of these cases of writing, the possibility that either the technology itself or the idea of it transferred between cultures cannot be excluded. The geographic and temporal proximity of Mesopotamia and Egypt make technological and conceptual influence not only possible but likely, and though Chinese writing is more distant in both space and time, the idea that writing could have spread to China from the first two is not farfetched. But unlike writing, which requires specific invention, numbers emerge from a perceptual capacity shared by the entire human species; moreover, this capacity is embedded in social and environmental contexts potentializing its expression and elaboration in material, behavioral, and linguistic forms. Numbers do spread, as the Sumerian numbers demonstrate, but they tend not to encounter situations containing no numbers whatsoever; rather, they are likely to encounter other numbers, whose characteristics can influence how many of the new numbers are adopted and whether they’re adapted in the process. Like a single origin does, the several-origins hypothesis believes some societies to be more capable of inventing numbers than others. And some of the ideas proposed for numerical spreading rest on outdated assumptions, debatable evidence, and misunderstandings or mistakes, like the idea the Oceanic lan15 Lean, ‘Counting Systems of Papua New Guinea and Oceania’; Owens and Lean, History of Number: Evidence from Papua New Guinea and Oceania. 16 Owens and Lean, History of Number: Evidence from Papua New Guinea and Oceania, p. vii. 17 Comrie, ‘Typology of Numeral Systems’, p. 1.

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guages are Semitic18 or that Polynesian numbers reflect Mesoamerican influences.19 These hypotheses tend to be inconsistent with current archaeological, linguistic, and genetic estimations of ancient population movements and relations. These circumstances suggest a several-origins model similar to that envisioned for writing is less accurate in characterizing numerical origins than the multiple-origins model. Are written notations and precursor technologies like tokens essential to mathematical elaboration? I think they would have to be, though numbers can become highly elaborate without them. A good example is found in Polynesia, where counting behaviors involved multiple people or rearranged the enumerated objects themselves, like the Tongan ceremonial counting and Māori tally counting described in Chapter 5. These behaviors produced complex concepts of numbers with two-dimensional structure and relations much more elaborated than those of an ordinal sequence. But without notations—without the concision, entitivity, and persistence notations add to the cognitive system for numbers—would Polynesian numbers have become elaborated as a mathematical system? This seems unlikely. Numbers acquire the properties that make them numbers from the material forms used to represent and manipulate them, linking the material forms used with the properties numbers have. With counting behaviors that rearrange the goods themselves, it’s possible to realize relations between numbers to a degree analogous to those explicated with tokens, as the Polynesian numbers demonstrate. However, rearranged goods represent numerical information with a perishability comparable to that of the fingers: Sooner or later, they’ll be needed for a purpose other than instantiating numerical information. Using such ephemeral forms also means the power of material forms to encode, accumulate, and distribute numerical knowledge will not have been harnessed. And rearranged goods have a limited ability for the manipulation needed for explicating relational information to a degree reasonably construed as mathematical. Similarly, without the handwriting effects and concision notations provide, there’s no mechanism for realizing the entitativity or relational data that make numbers a relational system. Without the challenge of notations’ fixedness, there’s neither mechanism nor motivation for developing strategies and algorithms to manipulate numerical relations. Mathematical elaboration, then, requires an elaborational mechanism, the recruitment of multiple material forms and the endurance of their properties in implicit form; representational concision, which notations have but their precursors lack; and the ability to accumulate and distribute cognitive effort, something material counting devices do with a high degree of specificity, fidelity, and persistence.

18

MacDonald, Oceania: Linguistic and Anthropological, p. 12; Thomas, ‘Maya and Malay’,

p. 92. 19 Heyerdahl, ‘Did Polynesian Culture Originate in America?’; criticized in Heine-Geldern, ‘Heyerdahl’s Hypothesis of Polynesian Origins: A Criticism’ and Schuhmacher, ‘On the Linguistic Aspect of Thor Heyerdahl’s Theory: The so-Called Non-Polynesian Number Names from Easter Island’; see related criticisms in Jakubowska, ‘The Spanish Expedition to Easter Island, 1770: Original Documents and Their Rendition by Bolton Glanvill Corney’.

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Does the increase in implicit knowledge required to use written notations ever make numbers a wholly mental phenomenon? I think not, since it’s difficult to conceive of number concepts without some kind of material form to make them tangible, intelligible, and manipulable. Consider what numbers are initially: judgments that sets of objects share quantity. This judgment draws upon qualities of the physical world like objects and the quantity they instantiate, as well as capabilities of the human brain for perceiving quantity and making relational judgments. From their inception, then, number concepts are distributed over external physical and internal mental domains. Consider too what numbers are at the point in their elaboration where they have become conceptual entities in a relational system with an organizing base number. If they require more knowledge in the internal mental domain than what the number sense provides, they nonetheless still require external physical representations like notations that scaffold their acquisition and structure and organize how they are understood and used. Thus, numbers remain materially bound, not just at their most elaborated but also no matter how many material devices they have acquired properties from and no matter how many devices they are distributed over and independent of. The question of whether and the degree to which numbers are in-the-head phenomena is a matter of ongoing debate.20 Currently, the answers seem black-and-white: Numbers are either entirely a matter of neural activity inside the brain, or they are not because they also involve extracranial assets like the body and material structures. Obviously, I take the latter view. While presumably internal capacities like the number sense, categorical judgments of relation, and language are essential to numbers, even these things cannot be said to be strictly mental phenomena: Both the perceptual experience of quantity and relational judgments of similarity and dissimilarity in quantity necessarily involve external material structures. Expressing numbers in language not only follows perceiving quantity and expressing quantity judgments in non-linguistic form, material combinations and properties also inform how number words are produced in language. The converse is also true, since presumably external structures like material devices, their properties, and the behaviors manipulating them assume an internal mental domain capable of supplying what they don’t explicitly represent and appreciating the resultant structure, organization, and patterns. Performing mathematical tasks also engages neural circuitry for physical movement (Chapter 4), making numbers inherently a system for manipulating material objects no matter how fully in-the-head they may seem. Substantial effort is being put toward understanding numbers, especially within psychology and linguistics. I won’t claim these fields should try to understand what material forms have to do with numbers merely to acquire knowledge for knowledge’s sake, though this would not be an entirely inappropriate basis for such inquiry. I do suggest considering material devices because their potential to influence psychological, behavioral, and linguistic aspects of numerical cognition far exceeds their role in histor-

20 Nieder, ‘Number Faculty Is Rooted in Our Biological Heritage’; Núñez, ‘Is There Really an Evolved Capacity for Number?’.

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ical elaboration.21 For example, cross-cultural research in numerical cognition tends to approach numbers as a monolithic construct. Yet numerical systems inhabit a spectrum in which some have highly elaborated relations and operations and are distributed over multiple material forms, while others lack relations, operations, and material distribution. When psychological reactions are compared across the spectrum, elaborational differences have the potential to affect the constructs being measured, the asymmetry I mentioned in Chapter 11 for 18th-century CE Spanish and Polynesian numbers. Spaniards looking at Polynesian numbers would have found a subset of familiar concepts, likely more intuitive in having a conceptual basis. Polynesians looking at Spanish numbers, on the other hand, would have encountered either concepts different from the ones they knew or entirely unknown to them. And if the Polynesians might have found the Spanish numbers somewhat difficult to understand, they were still more likely to have understood them than peoples with ordinal numbers or a few distributed exemplars of subitizable quantity, as these forms represent number systems in which the concepts differ from those of the Spanish to an even greater extent. Understanding their material origin also lets us see numbers as a mix of explicit material representation, on the one hand, and implicit knowledge and neurological reorganization on the other. The proportion of the two differs between notational systems,22 and between systems using various non-notational material forms. I suggest numerical content, structure, and organization—whether numbers are conceived as equivalences, collections, or entities; whether they have exponential dimensions and a base number; how they are constructed, and how many terms are productive; the amount of implicit knowledge or neurological reorganization a user must supply to understand their external representations—can be understood by examining the associated material forms and counting behaviors. The insight this would generate might have potential utility in matters like the ease with which number concepts are acquired, especially in regard to learning difficulties or performing mathematical tasks. If material forms are so essential to numbers, why don’t we notice they are, either when we use them or as a matter of historical emergence? Certainly, their role in this regard seems invisible much of the time. One reason for this invisibility is we tend to think of ourselves as agents and materiality as the instruments helping us achieve our goals, a characterization making materiality passive. An axe does not cut wood on its own; rather, a person adds the necessary movement and intent. Lacking any ability to move on its own or motivation to split wood or build a fence, the axe seems inert. Even while it is being used and is the most unambiguously integral to cognition, the axe neither understands its purpose nor has any capacity for such understanding, one of the key differences between human and material agency. But when we use an axe, it is integral to both the cognitive system and the physical outcome it helps produce: Each stroke of the axe is modified or corrected, according to the shape of the cut face of the tree left by the previous stroke. This self-corrective (i.e., mental) process is brought about by a total system, trees-eyes-brain-muscles-axe-stroke-tree; and it is 21 22

Overmann, ‘Constructing a Concept of Number’. Zhang and Norman, ‘A Representational Analysis of Numeration Systems’.

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this total system that has the characteristics of immanent mind. … The total selfcorrective unit which processes information, or, as I say, “thinks” and “acts” and “decides,” is a system whose boundaries do not at all coincide with the boundaries either of the body or of what is popularly called the “self” or “consciousness”.23

Such transparency to experience may have a neurological basis: Our neurons adapt to tool use, incorporating tools into our body schema or image: The neurons controlling finger movements react to tools as if they were part of the hand, allowing them to function as extended fingers.24 Visually impaired people exploit this phenomenon when they navigate with a cane, which extends their tactile perception to its tip.25 In a sighted person, using a stick or an implement like a rake additionally remaps visual space, such that things within the extended reach of the tool seem nearer.26 Interestingly, simply holding a tool does not appear to have this effect on visual nearness; rather, the tool must be put into action for the effect to occur.27 Much of our cognitive activity—acts of perception, forming and recalling of memories, learning, information processing, and behavior—is unconscious in any case.28 That is, no conscious planning goes into producing movements or speech. We don’t deliberately think through how our arm and hand positions need to be changed from instant to instant to pick up an object, or the ways we need to arrange and integrate phonetic, morphemic, lexical, and syntactic elements to form coherent sentences. Rather, we simply move and talk. We do have some aware moments: We can produce or avoid specific actions or choose our words carefully to match performative expectations and contextual situations. Nonetheless, such awareness is monitoring at a high level and does not constitute access to the totality of planning and executing. There are situations in which we do seem to be more aware of our movements, like learning to drive. As proficiency is gained, the portion of conscious awareness dedicated to monitoring decreases, with the result the movements involved become automated. This automaticity frees up resources of attention for purposes like ‘[g]oals, motives, and selfregulation’.29 This effect too can make the use of material forms transparent to our experience. All these effects—how we think about human agency and intentionality, the incorporation of tools into our body schema, our general lack of conscious access to the planning and executing that goes into movement and speech, and effects of automaticiBateson, Steps to an Ecology of Mind, pp. 317–319. Vaesen, ‘The Cognitive Bases of Human Tool Use’. 25 Malafouris, ‘Beads for a Plastic Mind: The “Blind Man’s Stick” (BMS) Hypothesis and the Active Nature of Material Culture’; Merleau-Ponty, Phenomenology of Perception. 26 Maravita and Iriki, ‘Tools for the Body (Schema)’. 27 Maravita et al., ‘Reaching with a Tool Extends Visual-Tactile Interactions into Far Space: Evidence from Cross-Modal Extinction’; ‘Tool-Use Changes Multimodal Spatial Interactions between Vision and Touch in Normal Humans’; ‘Active Tool Use with the Contralesional Hand Can Reduce Cross-Modal Extinction of Touch on That Hand’. 28 Kihlstrom, ‘The Cognitive Unconscious’; ‘Cognition, Unconscious Processes’. 29 Uleman, ‘Introduction: Becoming Aware of the New Unconscious’, p. 6. 23 24

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ty like attentional refocus—make materiality experientially transparent as a component of our personal cognitive machinery, so we are less aware of moving the pencil to write numerical signs than we are of making mental judgments in calculating. We also tend to take for granted the qualities and abilities that make materiality a mechanism for the co-evolution of human cognition and culture30: forming, anchoring, stabilizing,31 representing, and manipulating our concepts; accumulating our knowledge and distributing our cognitive effort over space and time32; and accumulating and recreating change in our behaviors and psychological processing.33 Both materiality’s change in form and our change from interacting with it occur on time scales outside our experiential grasp. Change in the form of writing, for example, was incremental and required multiple generations, thus exceeding the notice and lifespan of any single individual. Specific changes made by individuals would have seemed to be just that, not systemic change in writing itself. When we learn to read and write, there does not seem to be a single memorable moment to mark the before and after of attaining skill; rather, the process of learning and neurological reorganization that constitute proficiency is distributed and accumulated over months, if not years. Interestingly, when I consider Mesopotamian calculations, the material form is foregrounded to my experience in a way it is not when I calculate using the methods I learned in school. For example, problem 1, obverse column I, lines 1–4 of BM 13901, a 2nd-millennium BCE mathematical tablet, has been translated as, J’ai additionné la surface et (le côté de) mon carré: 45´. Tu poseras 1°, l’unité. Tu fractionneras en deux 1°: 30´. Tu multiplieras (entre eux) [30´] et 30´: 15´. Tu ajouteras 15´ à 45´: 1°. 1° est le carré de 1°. 30´, que tu as multiplié (avec lui-même), de 1° tu soustrairas: 30´ est le (côté du) carré.34 [I added the surface and (the side of) my square: 45´. You shall set 1°, the unity. You divide in two 1°: 30´. You multiply (them) [30´] and 30´: 15´. You add 15´ to 45´: 1°. 1° is the square of 1°. 30´, which you have multiplied (with itself), from 1° you subtract: 30´ is the (side of) the square.]

As x2 + x = 45´ is given, the scribe calculates the area as x = –30´ + √(30´2 + 45´) = 30´.35 The lines have been explained as follows:

30 Overmann and Wynn, ‘Materiality and Human Cognition’; ‘On Tools Making Minds: An Archaeological Perspective on Human Cognitive Evolution’. 31 Hutchins, ‘Material Anchors for Conceptual Blends’. 32 Hutchins, Cognition in the Wild; Smith, ‘Object Artifacts, Image Artifacts and Conceptual Artifacts: Beyond the Object into the Event’. 33 Malafouris, How Things Shape the Mind: A Theory of Material Engagement; Overmann, ‘Thinking Materially: Cognition as Extended and Enacted’. 34 Thureau-Dangin, ‘L’équation du Deuxième Degré dans la Mathématique Babylonienne: D’après Une Tablette Inédite du British Museum’, p. 31; also see Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin, p. 11. 35 Thureau-Dangin, p. 31.

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The square □(s) represents the surface; from this a projecting line 1 is drawn (“posited”); together with the side this “projection” contains a rectangle ٌٍ(1, s), whose surface is evidently equal to the side s. According to line 1, the total surface of square and rectangle is thus 45´. “Breaking” the “projection” into two parts 30´ and 30´ and making these “hold” each other as sides of a rectangle (indeed a square) projects a completed gnomon, whose surface is 45´ + 30´ × 30´ = 45´ + 15´ = 1, which is flanked by the “equalside” 1. “Tearing out” that 30´ which was moved around in order to “hold” leaves 1 – 30´ = 30´ as the (vertical) side s of the original square.36

The technique used is ‘based on the transformation of squares and rectangles with measurable sides by means of (1) cut-and-paste procedures, often in combination with (2) proportional stretching in one direction’.37 It has a sense of manipulating a material form—cutting a rectangle into pieces and rearranging the pieces as a square—that seems absent from procedures like multiplying length times width or calculating the quadratic function ax2 + bx + c. There is a true difference between the two techniques, since the Mesopotamian calculation did involve moving rectangular pieces, while the modern calculation does not but manipulates relations between numbers and variables instead. Part of it is perspectival, the difference between thinking through a material form, with all its experiential transparency, and thinking about one as an object of regard.38 There may also be a difference in the way the number concepts are distributed over multiple material forms, with the Mesopotamian calculation involving fewer material forms and thus less distribution and independence from them, the modern calculation more forms, more distribution, and more independence. Calculations documented in narrative texts, artifacts like tokens and tallies, and the behaviors these imply are what modern scholars have to reconstruct ancient numeracy. Time has removed them from their psychological and conceptual dimensions, which we must infer from what we know of such things from modern study. Without their psychological and conceptual dimensions, cognition approached through its material and behavioral components can appear mechanical, and hence, concrete. I assume, however, for someone inside the cognitive system with full access to its psychological and conceptual dimensions, the materiality was likely as transparent to its ancient users as ours is today.

DIRECTIONS FOR FUTURE RESEARCH In my current postdoctoral research, a project generously funded by the European Union Horizon 2020 program and hosted by the University of Bergen, I am comparing the multiple counting systems of Mesopotamia to the multiple counting systems of Oceania. Beyond this shared characteristic, what piqued my interest was that both systems had been described as examples of concrete, non-abstract thinking. I hope the comparison will generate new insights into why societies count various types of objects Høyrup, Lengths, Widths, Surfaces, p. 14. Høyrup, ‘Geometry, Mesopotamian’, p. 1. 38 Overmann and Wynn, ‘Materiality and Human Cognition’, pp. 7–8. 36 37

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with different numbers, perhaps explaining why peoples count in this manner and further supporting the idea such counting does not represent a particularly concrete or fractured notion of number. This is only one of many potential directions for the research presented in this volume. Another would be confirming whether my prediction that notations imply precursor technologies like fingers and one- and two-dimensional devices holds as true for the number systems of Egypt, China, and Mesoamerica as it appears to have for Mesopotamia, and whether differences among the devices used can account for the variability between these systems. Expanding my model of literacy is also on my list of things to do, with two potential directions. First, the development of literacy doesn’t seem to exhibit quite the same degree of universality as the development of numeracy does. Writing developed in Egypt at about the same time as it did in Mesopotamia, but Egyptian writing appears to have experienced not one but two phases of standardization and losing depictiveness and detail. Possibly this reflects a difference in social use, a factor in the amount of writing behavior and thus the speed of change: Where Mesopotamian writing developed from administrative practices, Egyptian writing was more aligned with religious and state purposes. Incorporating cases like Egypt into the model of literacy will help gain traction on the critical changes, temporal sequencing, and functional interdependencies inherent in the process. The second direction for expanding the literacy model is in regard to change in numerical notations over time. I see subitizing and counting as factors in the way notational forms conserve or simplify to avoid counting, and it would be interesting to examine how this works at a higher level of detail and resolution. Viewing numbers as a polylithic construct has potential implications for contemporary research in numerical cognition. For cross-cultural research, the asymmetric perspectives of more-elaborated-looking-at-less-elaborated and less-elaboratedlooking-at-more-elaborated may affect the constructs being measured and compared. For mathematical education, especially research into various difficulties affecting the acquisition and use of numerical concepts, viewing notations as mixes of explicit representation and an implicit component involving knowledge and neurological reorganization may suggest new teaching approaches or mitigation strategies. For linguistics, the various ways in which numerical language differs from language generally may suggest new ways of thinking about numerical language, perhaps a diachronic view to supplement the dominant synchronic view of linguistic variation or research into the mechanism(s) whereby language absorbs and echoes material properties like ten-ness in verbal productivity. Perhaps too there’s something to explore for language origins in the idea of the implicit knowledge needed to use material forms, for after all, sound is a physical phenomenon, ‘with words being a special case’ of the material forms we think through.39 For mathematics, understanding how numbers originate and elaborate might free them from their material roots. Just as non-Euclidean geometry once discarded assumptions gained from the experience of physical angles, shapes, and relations, the 39

Roepstorff, ‘Things to Think with: Words and Objects as Material Symbols’, p. 2051.

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Kakoli numbers described in Chapter 5 might someday confront the plodding conventionality of the successor function. What might numbers look like once the straitjacket of received wisdom has been loosed? A glimpse is provided by mathematician Brian Rotman, who suggested a similar deconstruction of the number line: Transiterates, numbers so large that counting them would exceed the universe’s resources, might be envisioned as escaping that straight-and-narrow path: Reconceived in non-Euclidean terms, transiterates would be free to ‘[float] off to the side somewhere, as if the number line fans out into a spray of disordered numbers’, a conceptualization with potential utility ‘for studying time’.40 While I have described the Ancient Near East as a situation where incorporating additional material devices yielded an increase in the elaboration of number, there is no inherent directivity to the process. That is, numbers are not foreordained to move from a state of less elaboration to one of more; they will do so under certain sociocultural conditions, including increased population size, contact with other social groups, and pressure from the internal and external complexity these imply. Under different sociocultural conditions—perhaps decreased population size, greater isolation, and less complexity to manage—numbers might lose elaboration. In this scenario, what might happen to the implicit component of structure and knowledge needed to understand notational forms is a good question. It’s possible that if we no longer used numerical notations, the implicit component would gradually disappear. This matter seems tractable to empirical analysis, and thus, is a matter for potential future inquiry. This research provides a unique window into how societies of average people achieve complex systems like mathematics and literacy. Two distinct mechanisms have been presented: Numbers elaborate by recruiting new material forms, acquiring their properties, and becoming more concise in what they explicitly represent as these attributes pass into implicit form. In contrast, writing elaborates by changing its single form in ways improving its ability to elicit specific behaviors and neurological responses in its users. Together, these mechanisms are unlikely to exhaust the ways materiality participates in and influences our cognition and its evolutionary change. Certainly, a third mechanism appears likely in the skilled production of things like pottery, an interaction of brain, body, and material form placing a greater emphasis on sensory, nondeclarative thinking.41 Nonetheless, the insights generated may be applicable beyond investigating how such complex cultural systems developed in ancient times, as they may bear upon the co-evolution of cognition and culture in general, including our own present and future. Finally, this research highlights the critical importance of the Ancient Near Eastern cultural heritage and the relevance of its continued study, making imperative the need to find solutions to the current regrettable state of abandonment, looting, and destruction. Solutions will hopefully be found before this precious world heritage has irrevocably vanished.

40 41

Rotman, Mathematics as Sign: Writing, Imagining, Counting, pp. 133–135. Malafouris, ‘At the Potter’s Wheel: An Argument for Material Agency’.

APPENDIX: DATA TABLES Pre-Uruk V (8500–3500 BCE) numerical impressions and tokens Source

Uruk

Umma

Artifact W 20987,11 P274833 W 20987,16 P274835 W 20987,07 P274843 W 20987,08 P285630 W 20987,03 P285629 W 20987,09 P285631 W 20987,12 P285632 W 20987,13 P285633 W 20987,15 P285634 W 20987,17 P285635 W 20987,18 P285636 W 20987,20 P285637 W 20987,22 P285638 MS 4631 P235737 MS 4632 P235738 MS 4633 P235739 MS 4634 P235740 MS 4635 P235741 MS 4636 P235742 MS 4637 P235743 MS 4638 P235744 MS 4646 P235745

Type Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens

N64

N45 2N45 2 SP

N34

N14 2 N14 2 sp 5 N14 5 sp

N1

N39 2 N39 2 di

N41

N3

Other

7 N3 7 ov 5 N64 5 TE

2 N39 2 di 1 N39 1 di 4 N39 4 di 5 N39 5 di

1 N14 1 sp

2 SP 1 N45 1 SP 1 N45 1 co 1 N45 1 SP 1 N45 1 SP

1 N34 1 CO 2 N34

5 N34 5 co

4 N14 4 sp 4 N14 4 sp 4 N14 4 sp 1 sp

4 N14 1 sp 7 N14 7 sp 5 N14 1 sp 4 N14 4 sp 7 N14 1LG 6sm 6 N14 6 sp

1 N45 1? LG 1 N45 2? LG 1 N45 1 SP

1 N1 1 cy 1 N1

1 sq

1 N39 1 di 3 N39 1 di 1 N39 1 di 1 N39 3 di 3 N39 3 di

2 N1 1 cy 3 N1 3 cy

2 lens

4 N41 4 te

4 tr

4 te 3 N1 3 cy 8 N1 3 cy 2 N1 2 cy

3 N39 2 di

6 te

4? sm

245

246

THE MATERIAL ORIGIN OF NUMBERS

Source

Artifact Sb 01927 P274841 Sb 06350 P281696 Sb 01940 P285645 NIM — P285654 NIM — P285655 NIM — P285657 NIM — P285658 NIM — P285659 Uncertain P285661 NMSDeZ — P283935 NMSDeZ — P285660

Susa

Tepe Yahya Habuba Kabira

Type Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens Bulla Tokens

N64

N45

N34

1 TE 3 N45 3 co

N14 4 N14 2 sp 3 N14 3 sp 3 N14 3 sp

1 N45

1 N34 1 co

3 N14 3 sp 4 N14 4 sp 2 N14 2 sp

N1 3 N1 2 cy 1 N1

N39

N41

N3

Other 2 bow ties 1 CO

3 N1 3 cy 6 N1 6 cy 7 N1 7 cy 2 N1 2 cy 5 N1 5 cy 1 N1 1 cy

1 CO 6 N3 6 ov

11 N1

2?

Table A.1. Pre-Uruk V (8500–3500 BCE) numerical impressions and tokens. The artifacts (n = 33) were found in the cities of Uruk (n = 13) and Umma (n = 9), Iraq; Susa (n = 8) and Tepe Yahya (n = 1), Iran; and Habuba Kabira, Syria (n = 2). The numbers of impressions on bullae surfaces and associated tokens are indicated, along with their interpretation: for example, N01 and cy (cylinder). Tokens other than the plain shapes used for numbers are listed as Other. Analysis of the data is presented in Chapter 9. The data were compiled between August 2014 and November 2015 from Amiet, ‘Approche Physique de la Compatabilité à l’Époque d’Uruk les Bulles-Enveloppes de Suse’, Mémoires de la Délégation Archéologique en Iran, Tome XLIII, Mission de Susiane; Damerow and Meinzer, ‘Computertomografische Untersuchung Ungeöffneter Archaischer Tonkugeln aus Uruk, W 20987,9, W 20987,11 und W 20987,12’; Le Brun and Vallat, ‘L’origine de l’Écriture à Suse’; Schmandt-Besserat, Before Writing: From Counting to Cuneiform; and the database of the Cuneiform Digital Library. Although Damerow thought the ‘combinations of differently shaped tokens in the [bullae did] not show regularities that would indicate the representation of standardized numerical systems’,1 there is an overall similarity and conformance to the conventions of the counting systems he and his colleagues published in 1993.2 The correspondences on W 20987,18 from Uruk (P285636) and NMSDeZ—(P285660) from Habuba Kabira did not conform to the general pattern, suggesting possible regional or temporal variations, the use of personal codes, afterthoughts, or mistakes. The data from Umma and Susa also show an interesting variation in representing N45 with both cones and large spheres, possibly because similar spherical impressions might be produced with a sphere and the base of a cone. Key: co, cone; cy, cylinder; di, disk; ov, oval; sp, sphere; sq, square; te, tetra; tr, triangle. Size, large or small, is indicated with capital or small letters. QuesDamerow, ‘The Origins of Writing and Arithmetic’, p. 159. Nissen, Damerow, and Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. 1 2

APPENDIX

247

tion marks indicate tokens of undetermined shape; their size is indicated as large or small: LG or sm. Numbers of impressions or tokens highlighted in bold (e.g., NMSDeZ—P285660 from Habuba Kabira) exceeded the expected bundling.

248

THE MATERIAL ORIGIN OF NUMBERS Uruk V (3500–3350 BCE) numerical impressions Source

Uruk

Artifact

Face Obv Col 1 Obv Col 2 Obv Col 3 VAT 15337 Bot Col 1 Bot Col 2 Rev Col 1 Rev Col 2 Obv Col 1 Obv Col 2 VAT 15360 Bot Col 1 Rev Col 1 VA 13624 Obv Col 1 Obv Col 1 VAT 17814 Obv Col 2 Obv Col 1 VAT 17811 Obv Col 2 Bot Col 1 VAT 17830 Obv Col 1 Obv Col 1 Obv Col 2 VAT 17818 Rev Col 1 Rev Col 2 Obv Col 1 VAT 21356 Rev Col 1 VAT 17835 Obv Col 2 VAT 21376 Obv Col 1 Obv Col 1 VAT 15299 Obv Col 2 VAT 17819 Obv Col 1 Obv Col 1 VAT 17815 Obv Col 2 VAT 15085 Obv Col 1 VAT 17816 Obv Col 1 VAT 15277 Obv Col 1 Obv Col 1 VAT 17817 Obv Col 2 Obv Col 1 VA 13629 Rev Col 1 W 06883,a Obv Col 1 W 06883,e Obv Col 1 VA 13622 Obv Col 1 VAT 17822 Obv Col 1 VAT 15332 Obv Col 1 Obv Col 1 W 06883,m Rev Col 1 W 06883,? Obv Col 1 W 06883,t2 Obv Col 1 Obv Col 1 VAT 15261 Obv Col 2 W 07067,a Obv Col 1 W 07067,c Obv Col 1 VA 13625 Obv Col 1 VA 13631(a) Obv Col 1 VA 13631(b) Obv Col 1 Obv Col 1 VAT 17854 Bot Col 1 Obv Col 1 VAT 15358 Obv Col 2 VAT 17849 Obv Col 1

N45 N48 N34 N15 N14 N1 N39 N24 N22 N23 N51 Other 1 9 5 3 3 1 6 5 2 2 2 1 4 4 4 1 3 2 1 3 5 5 1 2 2 3 2 2 2 1 4 2 3 2 5 DUG~b 3 2 3 2 1 1 4 UDU~a 9 UDU~a 1 4 3 2 2 5 2 1 2 2 2 2 3 1 2 1 2 1 7 1 4 1 2 1 3 5 2 3 2 1 2

APPENDIX Source

Artifact VAT 17841 VA 13632 VAT 15359 VAT 17812

VAT 17825

VAT 17826

VAT 17810 VAT 17836 VAT 17853 Uruk

VAT 17848

VAT 17813

VAT 17846 VAT 17820 VAT 21369 VAT 15355 VAT 15278 W 09656,ea+ W 09565,eb VA 13633 VAT 21288 VAT 17821 IM 134888 1851-01-01,0217 MS 5051 MS 4647 Tell Jokha Tutub Nineveh Jemdet Nasr

MS 4649 MS 4648 MS 3053 OIM A21310 (P235769) IM 132906 Ashm 19260634

Face Obv Col 1 Obv Col 2 Bot Col 1 Obv Col 1 Bot Col 1 Obv Col 1 Obv Col 2 Obv Col 1.1 Obv Col 1.2 Obv Col 2 Obv Col 3 Rev Col 1 Obv Col 1.1 Obv Col 1.2 Obv Col 2.1 Obv Col 2.2 Bot Col 1 Bot Col 2 Obv Col 1 Obv Col 2 Obv Col 1 Obv Col 1 Obv Col 2 Obv Col 3 Obv Col 1 Obv Col 1 Bot Col 1 Bot Col 2 Bot Col 3.1 Bot Col 3.2 Rev Col 1 Rev Col 2 Obv Col 1 Obv Col 2 Obv Col 1 Obv Col 1 Obv Col 2 Bot Col 1 Rev Col 2 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Rev Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 2 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Obv Col 2

N45 N48 N34 N15 N14 N1 N39 N24 N22 N23 N51 1 3 4 1 1 2 5 4 4 3 2 2 2 4 4 2 7 2 2 2 4 5 1 5 3 2 1 2 8 2 4 1 2 6 2 2 5 1 1 6 7 5 2 3 1 1 6 1 X 2 3 4 3 3 3 3 4 6 1 12 7 2 2 1 5 3 8 5 3

249 Other

250

THE MATERIAL ORIGIN OF NUMBERS Source

Susa

Artifact Sb 06313 Sb 22218 Sb 22225

Face Rev Col 1 Obv Col 1 Obv Col 1 Obv Col 1 Sb 22250 Obv Col 2 Sb 22262 Obv Col 1 (P008280) Obv Col 1 Sb 06961 Obv Col 1 Sb 04844 Obv Col 1 Sb 04854 Obv Col 1 Obv Col 1 Sb 22337 Obv Col 2 Obv Col 3 Obv Col 1 Sb 06369 Rev Col 1 Obv Col 1 Sb 06293 Rev Col 1 Top Sb 06299 Obv Col 1 Obv Col 1 Sb 06314 Rev Col 1 Obv Col 2 Sb 22371 Obv Col 3 Obv Col 1 Sb 06367 Obv Col 2 Obv Col 1 Sb 22426 Obv Col 2 Sb 06372 Obv Col 1 Obv Col 1a Obv Col 1b (P008553) Obv Col 2a Obv Col 2b Obv Col 1a Obv Col 1b Sb 22516 Obv Col 2a Obv Col 2b (P008632) Obv Col 1 Obv Col 1a Obv Col 1b (P008673) Obv Col 2a Obv Col 2b Obv Col 1 Sb 22624 Obv Col 2 Rev Col 1 Obv Col 1 T 084 Obv Col 2 (P008962) Obv Col 2 (P009093) Obv Col 1 (P009170) Obv Col 1 (P009171) Obv Col 1 (P009173) Obv Col 1 Sb 15284 Obv Col 1 Obv Col 1a Obv Col 1a Sb 15287 Obv Col 2a Obv Col 2b Sb 15248 Obv Col 1 Sb 15390 Obv Col 1 Obv Col 1 Sb 15436 Rev Col 1 S.ACR 1090.4 Obv Col 1

N45 N48 N34 N15 N14 N1 N39 N24 N22 N23 N51 Other 6 2 8 5 2 2 6 7 8 9 1 2 3 4 4 4 4 X 4 3 M072 1 4 4 1 1 1 8 5 1 7 2 1 4 5 4 2 4 3 14 14 6 4 6 6 6 3 2 5 9 9 2 5 1 1 4 2 3 5 2 4 4 |M351+1(N14)| 2 3 1 3 M017 1 1 2 4 5 2 1 4 1 4 5 5 2 8 3 2 M157~a 5 X 3

APPENDIX Source

Susa

Chogha Mish

Godin Tepe

Artifact Face S.ACR 1087.4 Obv Col 1 Obv Col 1 S.ACR 1097.4 Rev Col 1 S.ACR 1097.2 Obv Col 1 S.ACR 1200.2 Obv Col 1 S.ACR 1133.1 Obv Col 1 S.ACR 1593.1 Obv Col 1 S.ACR 1593.3 Obv Col 1 S.ACR 1677.1 Obv Col 1 S.ACR 1678.1 Obv Col 1 S.ACR 1593.2 Obv Col 1 S.ACR 1831.1 Obv Col 1 MT 758 Obv Col 1 Sb 06309 Obv Col 1 Sb 01976bis Obv Col 1 Sb 02316 Obv Col 1 Sb 06288 Obv Col 1 Sb 02312 Obv Col 1 Sb 04851 Obv Col 1 Sb 06291 Obv Col 1 Sb 02315 Obv Col 1 Sb 06289 Obv Col 1 (P281713) Obv Col 1 Sb 05874 Obv Col 1 Sb 01973bis Obv Col 1 MT 759(30) Obv Col 1 MT 759(27) Obv Col 1 Sb 04849 Obv Col 1 Sb 02317 Obv Col 1 MT 759(29) Obv Col 1 Sb 01975bis Obv Col 1 Sb 06859 Obv Col 1 MT 759(28) Obv Col 1 Sb 01980bis Obv Col 1 Sb 01944bis + Obv Col 1 Sb 03048 Sb 01987bis Obv Col 1 Sb 02276 Obv Col 1 Sb 02313 Obv Col 1 Obv Col 1 Obv Col 2 (P283936) Obv Col 3 Obv Col 4 Sb 06292 Obv Col 1 Sb 02135 Obv Col 1 Obv Col 1 II-432a-b Obv Col 2 Gd 73-054 Obv Col 1 (P009448) Obv Col 1 Gd 73-295 Obv Col 1 Gd 73-320 Obv Col 1 Obv Col 1 Gd 73-064 Obv Col 2 Obv Col 1 Gd 73-286 Rev Col 1 Obv Col 1 Gd 73-292 Rev Col 1 Gd 73-153 Obv Col 1 Gd 73-291 Obv Col 1 Gd 73-318 Obv Col 1

251

N45 N48 N34 N15 N14 N1 N39 N24 N22 N23 N51 Other 1 1 3 2 4 1 1 1 6 2 2 1 1 5 6 2 1 3 1 GADA~a@g 1 4 ZATU644 1 3 1 2 2 9 1 4 6 2 2 2 3 3 1 4 4 1 2 3 4 2 4 2 4 1 3 1 2

1 3

1 1

3

4 6 5 6 5 5 1 1 2 3 4 5 1 1 1 2 9

2

4

3 2 3 3 4 3 4 4

1

2

1

4 3 3 4 5 6

M009

252

THE MATERIAL ORIGIN OF NUMBERS Source

Artifact (Sialk 1617)

Face Obv Col 1 Obv Col 1 (Sialk 1619a) Rev Col 1 Obv Col 1 (Sialk 1621) Rev Col 1 Obv Col 1 (Sialk 1622) Obv Col 2 (P009515) Obv Col 1 Tepe Obv Col 1 Sialk (Sialk 1625) Rev Col 1 Obv Col 1 (Sialk 1629) Rev Col 1 Obv Col 1 (Sialk 1631) Obv Col 2 Obv Col 1 (Sialk 1632) Rev Col 1 (Sialk 1618) Rev Col 1 Ja 74-019 Obv Col 1 JA 74-020 Obv Col 1 JA 74-021 Obv Col 1 Ja 74-024 Obv Col 1 Ja 75-104 Obv Col 1 Jebel JA 75-105 Obv Col 1 Aruda JA 75-106 Obv Col 1 JA 75-107 Obv Col 1 JA 75-123 Obv Col 1 JA 78-526 Obv Col 1 (P249252) Obv Col 1 Mari T 084 Obv Col 1 Obv Col 1 (P235770) Obv Col 2 Obv Col 3 Habuba Obv Col 1 Kabira (P283931) Obv Col 1 Obv Col 2 (P283932) Obv Col 1 Tell Brak (P235766) Obv Col 1 Obv Col 1 MW 0188/001 Obv Col 2 Obv Col 1 MS 3140 Obv Col 2 Obv Col 1 MS 3141 Obv Col 2 MS 3142 Obv Col 1 MS 3143 Obv Col 1 MS 3144 Obv Col 1 Uncertain MS 3145 Obv Col 1 MS 3146 Obv Col 1 MS 3147/1 Obv Col 1 MS 3147/2 Obv Col 1 MS 3147/3 Obv Col 1 MS 3148 Obv Col 1 CUNES 51- Obv Col 1 02-003 Obv Col 2 Anonymous Obv Col 1 387697

N45 N48 N34 N15 N14 N1 N39 N24 N22 N23 N51 Other 2 5 1 2 1 X 1 M387 1 1 6 3 2 3 2 2 1 2 1 2 1 1 23 M072 1 6 1 1 2 7 5 10 3 1 2 4 1 2 7 2 22 5 3 10 5 5 5 1 4 1 5 11 3 1 4 3 1 3 4 1 1 4 2 1 1 3 2 2 3 1 4 8 9 6 2 2 1 5 4 5 2 4 4 4 1 9 2 2 1 3 3 1 6 3 5 [grain] 2 5 [grain] 4 4 2 1 7 3

4

Table A.2. Uruk V (3500–3350 BCE) numerical impressions. Numerical tablets (n = 193) from the cities of Uruk (n = 58), Tell Jokha (n = 5), Tutub (n = 1), Nineveh (n = 1), and Jemdet Nasr (n = 2), Iraq; Susa (n = 75), Chogha Mish (n = 1), Godin Tepe (n = 10), and Tepe Sialk (n = 10), Iran; Jebel Aruda (n = 11), Mari (n = 1), Habuba Kabira (n = 3), and Tell Brak (n = 1), Syria; and 14 whose provenience was uncertain. The data

APPENDIX

253

represent all the artifacts assigned to the Uruk V period with legible numerical signs in the database of the Cuneiform Digital Library as of the date when the data were drawn. Sign variants were omitted to simplify presentation of the data. Key: Obv, Obverse; Rev, Reverse; Bot, Bottom; Col, Column. When the database listed the museum designator for an artifact as uncertain, the database number was given instead. Numbers of impressions highlighted in bold (e.g., MS 4648 from Tell Jokha) exceeded the expected bundling.3 The data were drawn from the database of the Cuneiform Digital Library on 14 August 2015.

3

Nissen, Damerow, and Englund, Archaic Bookkeeping.

254

THE MATERIAL ORIGIN OF NUMBERS Newly catalogued tokens by site, reference(s), shape, and quantity Site

’Ain Ghazal, Palestine

Abu Hureyra, Syria

Abu Salabikh, Iraq

Amuq, Turkey Amuq, Tell Kurdu, Turkey Assur, Iraq Cafer Höyük, Turkey Çatalhöyük, Turkey

Chagha Sefid, Iran

Chogha Bonut, Iran

Tokens Cone Sphere Disk Cylinder Ovoid Miscellaneous Total Sphere Disk Ovoid Miscellaneous Total Cone Sphere Disk Cylinder Tetrahedra Ovoid Rectangle Triangle Miscellaneous Total Cone Sphere Disk Total Cone Disk Total Cone Sphere Tetrahedra Total Cone Sphere Total Sphere Disk Total Cone Sphere Disk Ovoid Miscellaneous Total Cone Sphere Disk Tetrahedra Ovoid Triangle Paraboloid OvalRhomboid Miscellaneous Total

Reference(s) 22 95 14 1 4 1 137 8 5 1 3 17 7 4 32 8 1 5 4 4 9 74 3 5 5 13 2 2 4 7 5 4 16 1 1 2 9 2 11 10 19 15 2 1 47 7 14 6 2 5 1 1 1 1 38

Iceland, ‘Token Finds at Pre-Pottery Neolithic ’Ain Ghazal: A Formal and Technological Analysis’, p. 50.

Moore and Tangye, ‘Stone and Other Artifacts’, pp. 173–177.

Baker et al., Abu Salabikh Excavations: The 6G Ash-Tip and Its Contents: Cultic and Administrative Discard from the Temple?, p. 133.

Yener et al., ‘The Amuq Valley Regional Project, 1995–1998’, pp. 202–209. Yener et al., ‘The Amuq Valley Regional Project, 1995–1998’, p. 204. Schmitt, Die Jüngeren Ischtar-Tempel und der NabûTempel in Assur: Architektur, Stratigraphie und Funde, p. 270. Cauvin et al., ‘The Pre-Pottery Site of Cafer Höyük’, p. 98. Hamilton and Uluceviz, ‘Figurines, Clay Balls, Small Finds and Burials’, p. 241.

Hole, Studies in the Archaeological History of the Deh Luran Plain: The Excavation of Chagha Sefid, pp. 233–237.

Alizadeh, Excavations at the Prehistoric Mound of Chogha Bonut, Khuzestan, Iran: Seasons 1976/77, 1977/78, and 1996, p. 86.

APPENDIX Site

Chogha Mish, Iran

Es-Sifya, Jordan Ganj-Dareh, Iran Hacınebi, Turkey

Jemdet Nasr, Iraq

Qal’at ’Ana, Iraq Sheikh-e Abad, Iran

Tall-iBakun, Iran

Tell ’Atij, Syria

Tell Abada, Iraq

Tokens Cone Sphere Disk Cylinder Tetrahedra Ovoid Rectangle Triangle Biconoid Paraboloid Animal Total Cone Sphere Disk Miscellaneous Total Cone Total Sphere Disk Total Cone Sphere Disk Tetrahedra Animal Miscellaneous Total Cone Total Cone Sphere Disk Cylinder Total Cone Sphere Disk Cylinder Tetrahedra Rectangle Triangle Paraboloid Miscellaneous Total Cone Sphere Disk Cylinder OvalRhomboid Miscellaneous Total Cone Sphere Disk Cylinder Miscellaneous Total

255 Reference(s)

10 13 17 1 10 4 1 8 1 3 2 70 36 38 1 3 78 644 644 10 2 12 40 12 4 2 1 2 61 1 1 1 3 2 5 11 17 6 16 2 1 1 2 5 3 53 8 3 5 1 1 3 21 64 84 15 9 8 180

Delougaz and Kantor, Chogha Mish: The First Five Seasons of Excavations 1961–1971, pp. 9A– 9B.

Mahasneh and Gebel, ‘Geometric Objects from LPPNB Es-Sifiya, Wadi Mujib, Jordan’, p. 107. Morales and Smith, ‘Gashed Clay Cones at Ganj Dareh, Iran’, p. 115. Stein, ‘The Uruk Expansion in Anatolia, p. 151; Stein et al., ‘Uruk Colonies and Anatolian Communities’, p. 230.

MacKay, Report on Excavations at Jemdet Nasr, Iraq; Matthews, Secrets of the Dark Mound: Jemdet Nasr 1926–1928, pp. 74-77. Northedge, Bamber, and Roaf, Excavations at ’Āna, p. 130. Cole, Matthews, and Richardson, ‘Material Networks of the Neolithic at Sheikh-e Abad: Objects of Bone, Stone and Clay’, p. 142.

Alizadeh, The Origins of State Organizations in Prehistoric Highland Fars, Southern Iran: Excavations at Tall-e Bakun, pp. 252–256; Langsdorff and McCown, Tall-i-Bakun A: Season of 1932, p. 68.

Weiss, ‘Archaeology in Syria’, p. 700.

Jasim, The Ubaid Period in Iraq: Recent Excavations in the Hamrin Region, pp. 69–73; Jasim and Oates, ‘Early Tokens and Tablets in Mesopotamia: New Information from Tell Abada and Tell Brak’, p. 355.

256

THE MATERIAL ORIGIN OF NUMBERS

Site Tell Brak, Syria Tell Madhhur, Iraq Tell Oueili, Iraq Tepe Abdul Hosein, Iran Zagheh, Iran

Ziyaret Tepe, Turkey

Tokens Cone Sphere Disk Rectangle Miscellaneous Total Cone Total Disk Total Cone Sphere Cylinder Total Cone Sphere Disk Tetrahedra Ovoid Rectangle Miscellaneous Total Cone Sphere Disk Cylinder Tetrahedra Rectangle Miscellaneous Total

Reference(s) 8 20 51 1 6 86 2 2 4 4 20 10 12 42 94 84 31 2 14 9 4 238 137 33 73 81 142 19 5 490

Oates, Oates, and McDonald, Excavations at Tell Brak: Nagar in the Third Millennium BC, p. 76.

Roaf, ‘’Ubaid Social Organization and Social Activities as Seen from Tell Madhhur’, p. 126. Breniquet, ‘Les Petits Objets de la Fouille de Tell El ’Oueili, 1983’, p. 142. Pullar, Tepe Abdul Hosein: A Neolithic Site in Western Iran: Excavations 1978, pp. 173–175.

Moghimi and Nashli, ‘An Archaeological Study on the Tokens of Tepe Zagheh, Qazvin Plain, Iran’.

MacGinnis et al., ‘Artefacts of Cognition: The Use of Clay Tokens in a Neo-Assyrian Provincial Administration’; Ziyaret Tepe Archaeological Project database.

Table A.3. Newly catalogued tokens. The newly catalogued tokens are listed by their provenience, the modern country where the site is located, and the source of the data. The new tokens reflected 13 of Schmandt-Besserat’s 16 categories4; most were the plain types used in numerical accounting: cones, spheres, and disks. Compiled from multiple sources.

4

Schmandt-Besserat, Before Writing: From Counting to Cuneiform.

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INDEX Abacus, 23 seq., 135, 171, 173, 235; compared to calculating with notations, 208; implicit/explicit knowledge in, 208 seq.; possible invention and use in Mesopotamia, 176 seq.; resemblance to ŠID, 176 seq. (See also Devices, two-dimensional; Game boards; Mental abacus.) Abipón numbers, 21, 75, 87, 94, 96. Abstract, 4 seq., 7, 25, 34; as loss of depictiveness in written characters, 56, 187 seq., 222 seq.; contextually defined, 56; in mathematics, 7, 27 seq., 35, 56, 222 seq. (See also Concrete; Properties, numerical; Properties, writing.) Abstract–concrete distinction, 2 seq.; arguments against, 201, 221 seq.; based on real phenomena, 66; change to more descriptive terminology, 66; in mathematics, 221 seq.; inherent bias in, 29 seq.; replacement, 219. (See also Mesopotamian exceptionality.) Abstraction, 53; behavioral, 55 seq.; material, 55 seq., 222 seq.; principle, in number systems, 137; psychological, 43, 62 seq., 223 seq.; representationalist views of, 35, 55 seq.; written forms, neurological reasons for, 190 seq. Acalculia, 17, 59. Additive inter-exponential organization, 202. Additivity, see Properties, numerical. Affordances, 5, 15 seq., 34, 41, 136; capacity of individuals and generations, 141; capacities and limitations of material forms, 209 seq.; hands and fingers,

134 seq., 209 seq., 236; encoded in devices, 12, 15; material and linguistic forms contrasted, 214; material devices used in Mesopotamian numbers, 210 seq.; tallies, 19, 145 seq., 211; tokens, 177 seq., 212. (See also Gibson, James J.; Material Engagement Theory.) Agency, see Material agency. Aguaruna numbers, 71, 87, 96. Aimoré numbers, 72. Ain el-Buhira tally, 147 seq., 151. Ainu numbers, 138 seq. Akkadian numbers, 109 seq., 216, 234 seq.; as oldest number system in Mesopotamia, 120 seq., 131; decimal structure in, 120, 132, 139, 216; fingercounting in, 139 seq.; grammatical number, 122 seq.; lexical numbers, 120 seq. Algorithms, 12 seq., 15, 135, 171, 198 seq., 208, 212 seq., 218, 226, 231, 236. Ambiguity, different representational modes, 186 seq.; in character form, 187 seq., 193; in character meaning, 185 seq.; instantiation in numbers, 185 seq.; tolerance for, in character form, 192. (See also Properties, numerical; Properties, writing; Script, cursive.) Amiet, Pierre, 1, 157 seq. Anatomically modern humans, 152. Andamanese numbers, 70, 75. Angular gyrus, 17, 58 seq., 218. Animacy, in grammatical number, 102, 121 seq.; in lexical numbers, 76, 102. (See

297

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also Grammatical number; Lexical numbers; Person.) Anthropology, historical emergence as discipline, 68. Anti-representationalism, 35; mental number line, 35 seq.; off-line states, 35 seq. (See also Change blindness; Representationalism.) Approximation, see Lexical numbers, fuzzy. Arabic numbers, 22, 103 seq., 111, 124, 126, 190. Archaeology, analysis, 30 seq.; analytical strategy, 133 seq.; challenges in considering non-archaeological evidence, 31; historical omission of fingers and notations from material analysis, 31; inferences from the material record, 133; invisibility of early material forms used for counting, 65, 109, 141; limitations and underestimations in, 109, 141, 151 seq.; preservation conditions in the Ancient Near East, 108 seq., 146 seq. Aristotle, arithmos, 223; essence of numbers, 25. Arrernte numbers, 76. Artifacts, as representational system, 35 seq. Assyriology, mandate to exclude crosscultural comparisons in, 105. Atomic terms, see Lexical numbers. Attention, freed to be repurposed, 217, 239; process governing subitizing, 16, 100, 103, 122; writing, 217. (See also Automaticity; Habituation; Object tracking; Working memory.) Australia, timeline for peopling, 46, 98 seq., 234. Automaticity, driving a car, 192 seq., 239; finger-counting, 138, 140, 217; proficiency and skill, 239 seq.; writing, 192 seq., 196, 217. Babylonian numbers, see Mesopotamian numbers.

Back-counting (or ‘backward counting’), 94, 200. Bakaïrí numbers, 70 seq., 85, 87, 94, 96, 118, 132. Behaviorally modern humans, 152. BM 13901, 2nd-millennium BCE mathematical tablet, 240 seq. Body-counting, 61, 69, 78, 83, 127, 153; arithmetical operations with, 27, 214; as expanded finger-counting, 61; in Papua New Guinea, 27, 77 seq., 153, 214. (See also Ordinality; Tallies; Yupno body-counting.) Brain region size, relation to adaptive use of brain functions, 58. Broca’s Area, 189. Broman, Vivian, 1, 157. Brouwer, Luitzen Egbeurtus Jan, 7, 28 seq. (See also Introspectionism; Intuitionism.) Bullae, 157, 159 seq., 165 seq., 180; noninvasive techniques with, 166; unbroken, 166 seq., 230. Bundling/debundling, 132, 161; expression of values in, 184 seq.; operations in, 163, 174 seq., 181 seq.; rules of exchange and replacement, 174 seq., 181 seq.; sources of relations, 175 seq.; tokens, 161, 204 seq.; unbundled maximum, 181 seq. (See also One-to-one correspondence.) Calculating, compared to counting, 208; continued use of tokens, 198 seq.; Mesopotamian, 240 seq.; new options related to notations, 198; notations, compared to an abacus, 208; visibility of materiality as function of familiarity, 200 seq. (See also Algorithms.) Cantonese numbers, 53, 102. Cardinal principle, in number systems, 137. Cardinality, 3, 25, 32 seq., 55, 137. Categorical judgments, identity, 53 seq.; relations, 53 seq.; relations across dimensions, 54; suppressing salient properties, 53 seq., 208.

INDEX Categories, of tokens, 160 seq. Categorization, 43, 53 seq., 194, 217; abilities of human and non-human species compared, 53 seq.; cross-cultural differences, 54 seq.; evolutionary reasons why humans outperform other species, 54 seq.; mental content in, 29; role of materiality in, 54 seq.; role of property sameness and difference, 53 seq. (See also Abstraction, psychological.) Cayönü Tepesi tally, 147 seq., 170. Cerebellum, role in creativity and numbers, 63 seq. Change blindness, 35. Change, in character form, 180, 190 seq. (See also Properties, writing.) Change, linguistic, in number words, 87. Change, numerical, 26 seq., 203 seq. Children, naïve experimenting with objects and language, 19 seq., 41; numerical acquisition, 52 seq.; numerical invention, 52. Chimpanzees, 12; categorization in, 53 seq.; symbolic numbers, 44 seq.; termite fishing, 12; working memory, 44 seq. Chinese numbers, 22, 105, 189 seq., 202 seq., 208, 215, 232 seq., 242. Chomsky, Noam, 46 seq., 67, 89, 92. (See also Discrete infinity; Generativity.) Chronology, material artifacts in Mesopotamian counting, 159 seq., 210; signs in Mesopotamian writing, 188. Chukchi numbers, 72 seq., 119. Ciphered forms, 202 seq., 208, 214, 224 seq., 233. Ciphered intra-exponential organization, 202. Coast Yuki numbers, 73 seq. Cognition, causality in, 10 seq.; computational model of, 48; conscious access to, 238 seq.; criteria for including materiality in, 13; extended and enactive, 6, 9 seq.; non-representationalist, 20; redrawing the boundaries, 10 seq., 24; traditional model of, 10 seq. (See also

299 Anti-representationalism; Conceptual blending; Material Engagement Theory; Representationalism.) Cognitive archaeology, requirements for analysis, 24. Cognitive technology, 99 seq., 219. Comparison, as mechanism for noticing property differences in numbers, 155. Complex cultural systems, 243. Complex tokens, see Tokens, complex; Tokens, plain vs. complex. Compounding, see Lexical numbers. Concept, defined, 132 seq.; experiential component of, 20; number, 45. (See also Conceptual blending.) Conceptual blending, 21, 36 seq., 133 seq., 203; behaviors, beliefs, expectations, habits, knowledge, and norms, 23, 38, 76, 133; expansion across individuals, at a time and across time, 41; Fauconnier and Turner’s original model, 36 seq.; Hutchins’ model with material anchor, 37 seq.; manuovisual engagement as locus of conceptual generation, 38; material anchoring, stabilizing, and storing, 19 seq., 37 seq., 54, 71, 92, 133 seq., 138, 221 seq., 240; social knowledge, 40. Conceptual integration, 21. Concision, 114, 198 seq., 203, 211 seq., 222 seq., 229, 231, 236; development across material sequence, 208 seq.; effects on consolidating representation, 208; effects on numerical relations, 198 seq.; increased requirement for implicit knowledge and neurological reorganization with, 208 seq. (See also Middle Eastern numbers; Properties, material.) Concrete, 7. (See also Abstract; Abstract– concrete distinction.) Concurrent use of material forms for counting, 213. Congruence, 215. Conjoined representation, see Tokens, conjoined representation.

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Conservation, see Numerical notations, conservation of form; Property contrasts. Contact, as elaboration mechanism, see Numerical elaboration. Context dependence, see Polyvalence; Tokens. Contrast, as elaboration mechanism, see Numerical elaboration. Contrastive, see Properties, writing. Convention, see Signification. Counting, compared to calculating, 208; by elevens, 82 seq.; collaborative, 82 seq.; with interstitial sticks, 73 seq., 84, 91; with the body, 27. Cumulative intra-exponential organization, 202. Cuneiform numbers, 22 seq., 112 seq., 120 seq., 158 seq., 197 seq., 202 seq., 208 seq. Cursive, see Script, cursive. Damerow, Peter, 2 seq., 50, 162 seq., 166, 219. (See also Abstract–concrete distinction; Numerical elaboration, timeline; Piaget, Jean.) Dano numbers, 214 seq. Demographic factors, 86 seq.; change and invention in Mesopotamia, 107 seq., 129 seq., 151 seq.; numerical elaboration, 88, 129 seq., 140. Depictiveness, see Abstract, as loss of depictiveness; Properties, writing. Descartes, René, 29. Determinatives, 187. Determinism, backward-looking, in numbers, 26; imposition of modern notions on archaic forms, 32, 220 seq. Devices, distributed exemplars of quantity, 21, 75 seq. Devices, one-dimensional, 78 seq., 211; beads and rosaries, 18, 30, 40, 56, 146, 152; beans and grain, 78, 174; gesture, 21, 38, 46, 63, 71, 75, 92, 131, 135, 215; leaves, torn, 77 seq., 141; pebbles, 19, 68, 78, 211; precursor technologies for Mesopotamian

counting, 154; strings, knotted, 18, 65, 77, 141, 174; stripes, marks, and dots, 76 seq., 92, 141, 145. (See also Body-counting; Finger-counting; Material devices; Tallies.) Devices, primary functions in numbers, see Manipulation; Representation. Devices, two-dimensional, 81 seq.; calculator, 12, 15, 26, 135, 141, 226; counting board, 176 seq., 235; quipu, 174. (See also Abacus; Finger-counting; Material devices; Mental abacus; Numerical impressions; Numerical notations; Tallies; Tokens.) Diachronic view, 69, 92, 242; historic ethnographic data, 69. (See also Synchronic view.) Differences, Mesopotamian and modern numbers and operations, 200 seq.; numerical notations and nonnumerical writing, 186 seq. Discrete infinity, 48, 91. Discreteness, influence of finger-counting, 17, 140; language, 91; numbers, 17, 91. (See also Properties, numerical.) Discrimination, of character forms, 191. (See also Individuation.) Distance effect, 44. (See also Number sense; Size effect.) Distinct means of accessing numerical intuitions, 92 seq. Distribution, 40, 219 seq., 225 seq., 241; concepts and multiple material forms, 201, 220. (See also Independence; Properties, numerical; Retention of older forms for counting.) Dolnì Věstonice tally, 144 seq. Double dissociation, 47, 196 seq.; neurofunctional independence of language and numeracy, 47 seq. Easter Island numbers, 207 seq., 215 seq., 220. Eblaite numbers, 115, 119 seq. Egyptian numbers, 129, 197, 232 seq., 235, 242.

INDEX Elaboration of numbers, see Numerical elaboration. Elamite numbers, 110, 121, 132, 205, 216, 234; decimal structure in, 121, 216; finger-counting in, 139 seq.; grammatical number, 122 seq.; known only in semasiographic form, 121; lexical numbers, 139 seq. Eme-sal, see Sumerian numbers. Enactive blend, 37 seq. (See also Conceptual blending.) Enactive cognition, 6. Enactive mind, see Cognition, extended and enactive. Enactive signification, 19 seq. (See also Material Engagement Theory.) Enactive space, 37. (See also Conceptual blending.) Encipherment as shift to implicit knowledge, 203. Enculturation, 52. English numbers, grammatical number, 100, 102, 121 seq.; lexical numbers, 22, 72, 95 seq., 215; ordinal numbers, 102 seq., 121 seq., 127. Erm 14645, Old Babylonian mathematical tablet, 200. Ethnographic data, historic, biases in, 68 seq.; caveats, 68 seq.; reasons for using, 69; unreliability of informants and observers, 68 seq. (See also Tongan numbers; Yupno body-counting.) Executive functions, 228; correlation with mathematical performance, 228. (See also Working memory.) Exner’s Area, 189, 192, 196 seq. Expression of numbers by iconic or indexical means, see Icon; Index. Extended mind, see Cognition, extended and enactive. External memory storage, 30 seq., 133; as passive repository of mental content, 31, 134. Familiarity, eyewitness identifications, 191; proficiency effects of, 208 seq.; writ-

301 ing, 191 seq. (See also Discrimination; Individuation.) Feature recognition, 190; local and global features of written objects, 190; optimization of local and global detail, 191 seq.; topological recognition of written objects, 190. (See also Fusiform gyrus; Visual Word Form Area.) Fijian numbers, 83. Finger agnosia, 17, 59. Finger gnosia, 47; predicting mathematical abilities, 59. Finger-counting, Akkadian, Elamite, and Sumerian, 139 seq.; attention and working memory, 61 seq., 217; conditions for numerical influence and patterning, 140; correspondences with counting in language, 138; crosscultural variability, 61 seq., 137 seq.; cross-cultural universality, 17, 62; demographic and elaborational span, 135, 140; dimensionality, 79 seq., 202; earliest materially structured counting, 4; effects on numbers, 140; exponential organization in, 202; fiveplus compounds in Sumerian, 132; hypothetical patterns, 137 seq.; impairments, 59; implications, 140; neurological basis, 17, 56, 134 seq.; nondeterministic nature of, 73; multiple individuals, 83; role in patterning numbers, 56 seq.; topographical organization of, 56, 60 seq., 136, 139, 213; traces in language, 40, 72 seq., 93, 132; typical patterning, 139; universal aspects, 137 seq.; variability, 73; why, how, and what, 133 seq. (See also Automaticity; Discreteness; Habituation.) Finger-montring, 136. Fingers, as bridge between internal and external domains, 17; capacity for and limitations in representing quantity, 209; device for representing and manipulating quantity, 17, 135 seq., 209; not significant in developing subtrac-

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THE MATERIAL ORIGIN OF NUMBERS

tion, 18; why used in numbers, 134 seq. (See also Affordances, hands and fingers; Hand.) Foundationalism, 28 seq.; Cartesian, 29; in mathematics, 28 seq. Frege, Gottlob, failure of foundationalist enterprise, 29; intersubjective verifiability of numbers, 7, 28. Frequency of use, 95; in lexical numbers, 95 seq.; in ordinal numbers, 103 seq. Fuegian numbers, 70. Functional fixedness, 213, 216; inhibiting effect on creativity and innovation, 216. (See also Materials shed.) Fusiform gyrus, 12, 189. (See also Visual Word Form Area.) Game boards, 176, 235. Gap numbers, 94. Generativity, 48, 91. Genetic data and ancient populations, 31, 108 seq., 155. Gibson, James J., 5, 15. (See also Affordances.) Globularization, evolutionary advantages of, 58. Glottographic, see Writing, glottographic. Gooniyandi numbers, 17. Grammatical number, 40, 53, 100 seq.; ancient languages, 121 seq.; aspects, 121 seq.; distribution in extant languages, 100 seq.; emergence from lexical numbers, 100 seq.; evidence of the number sense, 100, 121 seq.; first appearance in ancient texts, 113; frequency of use, 101 seq.; implications for priority of lexical numbers, 100; WEIRD/non-WEIRD distinction, 53. (See also Animacy; Lexical numbers; Number sense; Person.) Grammaticalization, 100; grammatical number, 100, 122; ordinal numbers, 103 seq. Greek numbers, 23, 49, 56, 107, 220 seq., 223 seq.

Grouping, counting devices, 80 seq.; fingers and toes, 56 seq., 80 seq., 138, 146; Mesopotamian tokens, 175; pairs and fours, 83 seq.; relations, 81; Sumerian numbers, 116; tallies, 16, 65, 81. (See also Bundling/debundling; Properties, material.) Habituation, finger-counting, 138, 140, 217; material forms, effect on creativity, 216; urban linearity, in WEIRD populations, 51 seq.; writing, 217. (See also Automaticity.) Hand, as actor and instrument, 60, 62, 135; bipolarity of, 60, 62; bridge between internal and external domains, 14, 60 seq.; compared to other species, 13 seq., 60; considered as material device, 4; proportion of dedicated motor and sensory cortex, 60 seq. (See also Affordances, hands and fingers.) Handwriting effects, see Writing, handwriting effects of. Hawaiian numbers, 215 seq., 221 seq. Ha-Yonim Cave tally, 146, 148 seq. Highest number counted, 218; approximation of numerical extent, 76, 87; effect on numerical elaboration, 218; unreliable as metric of numerical extent, 68. Hindi numbers, 22. Hindu–Arabic numerals, 22 seq., 40, 202 seq., 233. Homo species, 152; engagement of material forms by, 64; number sense, 44 seq. Homunculus, cortical, 60 seq.; Penfield’s original model, 60 seq. HS 201, late 3rd/early 2nd millennium BCE mathematical tablet, cover. Huchnom numbers, 73 seq. Icon, 21; expression of number by iconic or indexical means, 21, 38 seq., 75, 94; iconic use of sound, 75 seq. (See also Index; Signification.) Ideographs, 110, 184, 223.

INDEX Igbo numbers, 18. Implicit and explicit knowledge in numerical representation, 201 seq., 207 seq., 237; asymmetry of elaborational perspective, 209, 238; development in representational systems, 208; effects on concision and representation, 203. Impressions, see Numerical impressions. Incan numbers, 174. Inconspicuity, see Properties, writing. Incorporation of tools into body schema, 239. (See also Transparency.) Independence, 211, 219 seq., 225 seq., 241; concepts and material forms, 220 seq.; language and numbers, 21, 90; quality of numerical concepts, 56. (See also Distribution; Properties, numerical; Retention of older forms for counting.) Independently originated numerical traditions, see Akkadian numbers; Elamite numbers; Mesopotamia; Sumerian numbers. Index, 21; expression of number by iconic or indexical means, 21, 38 seq., 75, 94. (See also Icon; Signification.) Individuation, of character forms, 191. (See also Discrimination.) Indo-European numbers, 96 seq. Indonesian numbers, 234 seq. Inferential arguments, 5, 31. Instantiation, 22, 106, 180, 184 seq., 224; consequences in numbers, 22, 114; contiguity between notations and non-written precursors, 180. (See also Semasiographic notations; Signification.) Intentionality, see Material agency. Interaction, as elaboration mechanism, see Numerical elaboration. Interactivity, in (fixed) numerical notations, 135. Interdisciplinary research, inclusion of textual evidence, 4; inferential challenges, 5; use of data in analyses, 31.

303 Intra- and inter-exponential organization, 202. Intraparietal sulcus, 17, 57; distinguished in Homo sapiens and Neandertals, 57 seq.; evolutionary significance of, 57 seq.; neurological functions, 57. Introspectionism, 7; problems as basis for understanding mental content, 29. Intuitionism, 7, 28 seq., 35. Inuit numbers, Greenland, 72 seq., 82, 87, 96; Hudson Bay, 72. Invented principles of number systems, 137. Invention of writing, see Writing, invention of. Ishango bones, 143 seq. Isolate, linguistic, 65, 110. JA 74-024, Uruk V administrative tablet, 181. JA 75-104, Uruk V administrative tablet, 183. Jibaro numbers, 71, 80, 87, 96. Jita tally, 147 seq., 151. Judgments of identity, see Categorical judgments, identity. Juwal numbers, 118. Kakoli numbers, 84 seq., 242 seq. Kebara Cave tally, 146, 148 seq. Koryak numbers, 72. Ksar’Aqil tally, 147 seq., 150. Kwoma numbers, 77. Language, Akkadian, 110 seq., 155; bilingualism of Akkadian and Sumerian, 111 seq.; diacritics, in African tones and Hebrew vowels, 191 seq.; Eblaite, 115; Elamite, 110 seq.; evidence of material forms in numbers, 93; evidence of psychological capacities of ancient peoples, 93; evolutionary emergence of, 46 seq.; indiscriminate labeling of numerical concepts, 199; manuovisual means of accessing numerical intuitions, 64, 92 seq.; neural

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THE MATERIAL ORIGIN OF NUMBERS

activity, 48; physical phenomenon, 137; Proto-Afroasiatic, 128; ProtoSemitic, 128, 235; reasons for not excluding from numbers, 38 seq.; role in numbers, 39 seq., 46 seq., 89; Sumerian, 110 seq. (See also Chomsky, Noam; Double dissociation; Linearity.) Lanì numbers, 72. Latin numbers, 114 seq., 138, 202, 208, 233. Lebombo bone, 143 seq. Lesu numbers, 79. Levant, defined, 107. Level of counting, in Mesopotamia, 130. Lévy-Bruhl, Lucien, 50; theory of societal modes of thinking, 2 seq., 29, 50. (See also Piaget, Jean.) Lexical associations, see Writing, lexical associations. Lexical lists, 117, 230. Lexical numbers, 93 seq.; analyzable, geographic distribution of, 98 seq.; ancient languages, 115 seq., 139 seq.; arbitrariness, 19 seq.; atomic terms, 115; compounding, 94 seq.; compounding, in Sumerian lexical numbers, 115 seq., 231; compounding, relative age in numbers, 73 seq., 98 seq., 118 seq.; diagnostic characteristics, 87 seq., 104 seq.; etymological roots in fingers and material structures, 93; evidence of behaviors and material forms for counting, 87 seq.; features, geographic distribution in extant languages, 101; first phonetically specified forms in Sumerian, 113; fuzzy, in designating approximate numbers, 17, 27, 91 seq., 137; irregularity, 95; length, 87, 95 seq.; memorization effects in, 95; preceded and occasioned by use of material forms, 85 seq., 93, 116; phonetic representation in Sumerian numbers, 113 seq.; reasons why not phonetically specified in writing, 114; unanalyzability, 93, 98. (See also Akkadian numbers; Animacy; Elamite numbers;

Frequency of use; Sumerian numbers; Regularity.) Lexical rules, 48, 97 seq.; procedural memory system, 97 seq. (See also Mental lexicon.) Lexicalization, 97 seq. Lexicography: 195. Linearity, 17 seq., 37, 140; assumed in tokens prior to impressions, 140; contrasted in material and linguistic forms, 137; criterion for identifying tallies, 141; fingers and tallies, 17, 23; tokens and numerical notations, 23 seq. (See also Properties, material.) Linguistic change processes, 87, 95. Linguistic change, demographic effects in, 97 seq., 101 seq. Linguistic sign, 19, 92. Linguistic time depth of analysis, 30. Literacy, 179, 196, components, 179; conditions for producing, 190, 196; contrasted with numeracy, 196; emergence of, 188, 193 seq.; highly common but non-universal process, 242; reasons why numbers cannot be its basis, 196 seq. Local and global features, written objects, see Feature recognition. Locke, John, 89. Magnitude appreciation, 16, 44; threshold of noticeability, 16. (See also Number sense.) Magnitude ordering, 25; source of mathematical properties, 25; tokens, 176. (See also Properties, numerical; Ordinality; Russell, Bertrand.) Malleability, see Properties, writing. Mandarin numbers, 103 seq., 123. Manipulation, primary function of material devices in numbers, 30, 135; not all devices equally good at both manipulating and representing, 135. (See also Representation.) Māori numbers, 79 seq., 84, 236. Marquesan numbers, 83 seq. Marshall Islands numbers, 81.

INDEX Material agency, 14 seq.; agency defined, 14; compared to human agency, 14 seq., 238 seq.; displayed by changing human behavior, 14; effected through affordances, 15 seq.; influence on the emergence of arithmetical operations, 18; intentionality, 238 seq.; numerical cognition, 15 seq.; writing, 179. (See also Material Engagement Theory.) Material culture, as mechanism for accumulating and distributing cognitive effort, 24, 140 seq. (See also Demographic factors; Socio-material complexity.) Material devices, active contributions to human numeracy, 134; mechanism for elaborating numerical concepts, 175 seq.; two main functions of in numbers, 135; role in numbers, 86; use in representation and manipulation, 23; why used in counting, 85 seq., 133 seq. (See also Devices, onedimensional; Devices, twodimensional; Manipulation; Numerical elaboration; Representation.) Material Engagement Theory, 6, 9; three central commitments of, 9. (See also Affordances; Cognition, extended and enactive; Enactive signification; Linguistic sign; Material agency; Material sign; Materiality.) Material form of writing, see Writing, material form. Material forms, change, as representing change in behaviors and brains, 11, 180, 188 seq.; influence on language and numerical structure, 91; potential influence on the abstract–concrete distinction, 226 seq.; role in numerical origins, 90; Sumerian language, 116; use in distributing knowledge, 231. Material properties, as influence on writing, 193; as proxies for numerical properties, 5, 21, 55 seq. (See also Properties, numerical.)

305 Material sequence, in devices used in counting, 109; internal coherence of the devices used in Mesopotamia, 5, 209; key transitions of and conceptual change, 173, 211; Mesopotamian counting devices, 209 seq.; why devices are sequenced in counting, 86. Material sign, 92; differentiated from linguistic sign, 6, 19. (See also Enactive signification.) Material structure, consistency across forms and with language, 138; persistence, 37 seq., 40, 213; reasons for persisting across change in material forms, 213. Material structures, constitutivity vs. causal linkage in cognition, 6; mechanism of numerical elaboration, 5, 14, 209 seq.; role in anchoring and stabilizing concepts, 5, 19, 21, 133 seq., 221 seq.; role in cognition, 4; role in emerging numbers, 38. Materiality, broadly defined, 11; collaborative medium, 24, 40 seq.; constitutive of cognition, 10 seq.; instantiation of ancestral cognitive activity, 12; manipulation, role in realizing explicit concepts of numbers, 13; role in accumulating and distributing cognitive effort, 24, 140 seq., 231 seq., 236; role in decomposing complex tasks, 11 seq. (See also Affordances; Material Engagement Theory.) Materially influenced change in Ancient Near Eastern numbers, 217 seq. Materials shed, 216 seq. (See also Functional fixedness.) Mathematical, calculation, as enactive amalgam of thinking/scribbling, 6; elaboration, essentiality of writing and precursor technologies in, 236; thinking, as inherently alinguistic, 38, 89 seq. (See also Realism.) Mathematics, achievement in the Old Babylonian period, 3; freed from material origins, 242 seq.; importance of non-

306

THE MATERIAL ORIGIN OF NUMBERS

numerical writing, 197 seq.; theoretical vs. pure, 56, 224 seq. Maya numbers, 232 seq. Meaning, as brought forth through active engagement, 20; numerical, unambiguous, 157, 184 seq., 210; product of conceptual integration, 21. Mental abacus, 62 seq.; motor-movement planning, 63 seq. Mental activity, 36. Mental lexicon, 48, 95 seq. (See also Lexical rules.) Mental number line, 35 seq., 45; association with topographical structure in the brain, 62; challenge to antirepresentationalism, 36; debate over, 35 seq.; increased linearity influenced by material forms, 217. Mesoamerican numbers, 242. Mesopotamia, 107 seq.; case study, 4, 108 seq.; colonization, 107 seq.; connection with the Levant, 31, 108 seq., 155; defined, 107; ethnic and cultural identity, 108; independently originated number systems, 109 seq., 128, 234. (See also Akkadian numbers; Elamite numbers; Sumerian numbers.) Mesopotamian exceptionality, 3 seq., 66, 105, 109, 131, 163, 183, 225 seq.; arguments against, from ancient languages, 103 seq., 112 seq., 122 seq., 127 seq.; arguments against, from bundling relations, 163; arguments against, from demographic change, 129 seq.; arguments against, from dependence on material forms, 201; arguments against, from distribution of number over multiple material forms, 226; arguments against, from typical pace of invention, 154 seq.; historical development of the idea, 105; nonparsimonious nature of, 66; reasons for not assuming, 66 seq. (See also Abstract–concrete distinction; Representational separation.)

Mesopotamian numbers, 23, 56, 112 seq., 157 seq., 162, 202, 208, 220, 222 seq., 225 seq., 231, 233, 335. Mianmin numbers, 94. Models, conceptual blending, 21, 36 seq.; Russell’s logical types, as framework for categorizing in numbers, 33, 55. More alike, see Properties, writing. Motor cortex, 61; topographic organization of, 17 seq., 60 seq. Motor-movement planning and numbers, 47, 62 seq. Multiplicative intra-exponential organization, 202. Mundurukú numbers, 21, 27, 75 seq., 92, 131. Neolithic, defined, 108; revolution, 108. Neural activity in reading and writing, 189; differences in related to visual appearance of written characters, 189. Neural muscles, evolutionary development of, 64. Neuroanatomical differences between literacy and numeracy, 196 seq. Neuronal recycling, 13, 189. Non-independence in linguistic data, 86 seq. Nuer numbers, 76. Number appreciation, alinguistic species, see Quantity appreciation. Number concept, 45; defined, 25; defined to avoid referring to objects, 223 seq.; definition decomposed into analytical elements, 30; diagnostic characteristics in Mesopotamian languages, 87 seq., 104; differences between, 32; elaboration as relational system, 211, 218; emergence of, 3, 70 seq., 93 seq.; equivalences, collections, and relational entities, 197 seq., 211, 218 seq., 231; material history and prehistory, 31 seq.; monolithic vs. polylithic construct, 26, 238, 242; originating sequence, 93; reasons for borrowing in contact situations, 216; relational en-

INDEX tities, 211, 231; symbolic, appreciation, 44; system for containing information, 229 seq.; transiterates, 243; whether wholly mental phenomena, 7, 35, 231 seq., 237 seq.; working definition, 26; writing and numbers, as dissociable phenomena, 197. (See also Properties, numerical; Realism.) Number sense, 2, 15, 43 seq.; adaptive value of, 43 seq.; alinguistic species, 43 seq.; effects on cumulative notations, 203 seq.-5; effects on grammatical number, 100, 121; effects on lexical numbers, 93 seq.; effects on numerical emergence, 70; effects on ordinal numbers, 103 seq.; effects on language, 40; effects on numerical notations, 201 seq.; evolutionarily ancient perceptual system, 93; evolutionary emergence in the human lineage, 44 seq.; human infants, 47; imperviousness to the WEIRD/non-WEIRD distinction, 52; inherently alinguistic nature of, 47; perceptual experience of quantity, 15, 43 seq.; phylogenetic distribution of, 15, 43, 45 seq.; topographical structuring in, 60 seq. (See also Magnitude appreciation; Subitization.) Number words, see Grammatical number; Lexical numbers; Ordinal numbers. Numeracy, 47, 196; contrasted with literacy, 196; dissociation with language, 47; neural activity, 47. Numeral classifiers, 34. Numerals, see Numerical notations. Numerical elaboration, Ancient Near Eastern, 209, 219 seq.; asymmetry of perspective, 209, 238; conservation and change, 214 seq.; contact, 109 seq., 129, 154 seq.; contrast, 37 seq., 110, 213 seq., 216 seq.; defined, 219; interaction, 110; geographic distribution of, 86 seq.; likelihood when numbers are ingrained in culture, 219; nondirectional nature of, 99, 243; time-

307 line, 127, 138; universality, 231 seq. (See also Demographic factors; Material culture; Material devices; Material structures; Properties, numerical; Socio-material complexity; Writing, effects on numerical elaboration.) Numerical impressions, 110, 112, 180 seq., 245 seq.; correspondences with numerical notations and tokens, 5, 157 seq., 162, 180, 230 seq.; dimensionality, 202; ordered by increasing numerical value, 181; pre-Uruk V and tokens, 245 seq.; repetition quantities, 181; Uruk V, 248 seq. (See also Bundling/debundling.) Numerical invention, contributions of adults and children, 52; cross-cultural comparison of, 65, 137 seq.; species universality, 66. Numerical notations, 184; unambiguous numerical meaning of, 210; conservation of form, 22 seq., 203, 206; correspondences with impressions and tokens, 5, 157 seq., 162; differences with written non-numerical language, 196; interactivity, in (fixed) numerical notations, 135; essentiality for mathematical elaboration, 231, 236; fixedness, effect on calculating, 198 seq.; notational differences and precursor technologies, 242; organization of numerical notation systems, 202; phonetic independence, 21 seq., 114; semasiographic system, 21 seq., 112 seq.; universal characteristics, 231 seq. (See also Algorithms; Middle Eastern numbers; Typology, notational.) Numerical origins, problematic assumptions in, 33. (See also Origin hypotheses.) Numerical representation, basic types, 66, 186. Numerical tables, 198 seq., 202 seq., 223, 231, 236. Numerical tablets, geographic distribution of, 184.

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THE MATERIAL ORIGIN OF NUMBERS

Numerosity, see Number sense. Object recognition, see Feature recognition. Object tracking, 16, 100, 103, 122. (See also Attention; Automaticity; Habituation; Working memory.) Observer unreliability, see Ethnographic data. Occam’s razor, 66. Oceanian numbers, 70, 86, 235 seq., 241 seq. OIM A21310, Uruk V numerical tablet, 159. Oksapmin numbers, 27, 78 seq., 86, 92, 214. One, as number, 26; order of emergence, 70, 94. One-to-one correspondence, 34, 76 seq.; counting, 162 seq.; principle, in number systems, 137; tokens, 67, 162 seq., 177, 219; tallies, 79. Ontogenesis and enculturation, 52. Ontogenetic change, 12; development and numbers, 49 seq. Operations, accumulation, 18; addition, 18, 25; arithmetic, Ancient Near Eastern, 200 seq.; division, 19, 25, 41; exchange, 175 seq.; material forms, 222; multiplication, 25, 175 seq.; narrative descriptions, 200 seq.; reciprocals, 198, 200 seq.; remainder, 41; tokens, 175; typical order of emergence, 18, 146. (See also Bundling/debundling; Differences; Material agency; Semasiographic notations; Subtraction.) Oppenheim, A. Leo, 1, 157. Order-irrelevance principle, in number systems, 137. Ordinal frequency, 102 seq.; Sumerian, 125 seq. Ordinal numbers, 102 seq., 109 seq., 199; ancient languages, 113 seq., 123 seq.; correspondence with lexical numbers, 102; first appearance in texts, 113; frequency of use, 103 seq.; frequency

effects in, 124; implications for unrestricted numerical lexicons, 123, 127; representation in writing, 123 seq. (See also Lexical numbers; Number sense.) Ordinality, body-counting, 77 seq., 86, 92, 214; defined, 25; element of formal definition, 25; emergence of, 137; exchange operations in, 176 seq.; fingers and tallies, 78 seq., 132, 222; implications, 203; size-ordered sequence, 27, 78; relations, 78 seq., 236, 238. (See also Magnitude ordering.) Origin hypotheses, numbers, 231, 233 seq.; pertinent data in, 43; when counting began, 144; writing, 234 seq. (See also Numerical origins; Writing, invention of.) Orokaiva numbers, 80. Orthography: 195. P008553, Uruk V administrative tablet, 183. Pairing, 34; use in realizing a concept of three, 70 seq. Papua New Guinea numbers, 233 seq. Parietal, activity and numbers, 47; functions and tool use, 58. Pentadactyl limbs, see Finger-counting. Perception, color, 16, 90; distributed over external physical and internal mental domains, 237; dynamically interactive process, 10 seq., 20; neurological effects of tool use, 239; sound, 10, 16; threshold of noticeability, 16; touch, 16, 239; vision, 20. (See also Number sense.) Perceptual remapping, 239. Person, in grammatical number, 100. (See also Grammatical number.) Peruvian numbers, 174. Phonetic representation of Sumerian lexical numbers, 113 seq. (See also Writing, glottographic; Semasiographic notations.) Phonography, 111, 187.

INDEX Piaget, Jean, 2, 49 seq.; concern with how societies realize truth from opinion, 52; developmental theory, 50; developmental theory applied to Mesopotamian numbers, 105, 187 seq.; theory of societal modes of thinking, 2 seq., 29, 50. (See also Lévy-Bruhl, Lucien.) Pictographs, 110, 184, 223. Pirahã numbers, 27, 101. Place value, 201, 203, 216; metrological conversion, 223, 225 seq.; sexagesimal place value system, 201. (See also Contrast.) Plain tokens, see Tokens, plain vs. complex; Tokens, plain, used in numeration. Plato, 7; Platonic universals, 27 seq.; what makes numbers numbers, 25 seq. (See also Realism.) Polynesian numbers, 84 seq., 91, 235 seq., 238. Polyvalence, 161; context-dependence, 225 seq. (See also Properties, numerical.) Pomo numbers, 18, 75. Positional inter-exponential organization, 202. Postcentral gyrus, 61. Precentral gyrus, 61. Prime numbers, 7, 26. Procedural memory system, 97. Procedural memory system, see Lexical rules. Productive terms, 84, 95; in Polynesian numbers, 84 seq., 91; in Sumerian numbers, 116. Proficiency, as ontogenetic change, 11. Projection, 39. Properties, material, fixedness, 198 seq., 231; integrity of form, 19; manipulability, 18, 37, 175, 231. (See also Concision; Grouping; Linearity; Regularity; Transparency; Visual indistinguishability.) Properties, numerical, additivity, 25; entitivity, 197, 218 seq.; exactness, 91; intersubjective verifiability, 7, 28; irre-

309 ducibility, 221; materially bound, 219 seq., 227, 237; origins in different material devices, 40, 211; passing from explicit to implicit form, 212 seq.; passing from older to newer devices, 212; sequentiality, 17, 140. (See also Abstract; Ambiguity; Discreteness; Distribution; Independence; Magnitude ordering; Numerical elaboration; Polyvalence.) Properties, writing, contrastive, 191; depictiveness, 187 seq., 190, 223 seq.; inconspicuity of material forms, 179 seq.; malleability, 194; more alike, 191; simplicity, 191 seq.; standardization, 192 seq. (See also Abstract, as loss of depictiveness; Ambiguity; Transparency.) Property contrasts, introduced by new material forms for counting, 213 seq. Proto-Afroasiatic numbers, 128. Proto-cuneiform numbers, 112, 157 seq., 161 seq., 202, 208, 230 seq., 233. Proto-Germanic numbers, 97. Proto-Indo-European numbers, 97. Proto-language, reconstruction, 65; reconstruction of Proto-Indo-European, 96 seq. Proto-Semitic numbers, 128, 235. Psychological functioning in ancient peoples, challenges of accessing, 65. Quantical–numerical distinction, 45. Quantification behaviors, 78. Quantity appreciation, alinguistic species, 44, 208; contrasted with numeracy, 208. (See also Number sense; WEIRD/non-WEIRD distinction.) Quipu, 174. Reading, 179. Realism, 7, 27 seq., 35, 224 seq. Regularity, 18 seq.; criterion for identifying tallies, 141; rule-generated lexical numbers, 98; Sumerian numbers, 115 seq. (See also Properties, material.)

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THE MATERIAL ORIGIN OF NUMBERS

Relational judgments, see Categorical judgments, relations. Relations, declarative information, 218; elaborated, not pre-existing, 175; increased potential with higher numbers, 26; one-dimensional devices, 78 seq.; tallies, 145, 173 seq.; tokens, 67, 162 seq., 173 seq.; two-dimensional devices, 81 seq. (See also Middle Eastern numbers; One-to-one correspondence.) Repetition quantities, see Bundling/debundling; Numerical impressions. Representation, accuracy and persistence, 31; not all devices equally good at both representing and manipulating, 135; primary function of material devices in numbers, 30, 135; modes, 106, 184. (See also Manipulation.) Representational modes, see Signification. Representational separation, 177 seq., 201, 222 seq.; emergence of commodity labels, 184 seq.; relative to conceptual elaboration, 222 seq. (See also Ideographs; Pictographs; Tokens, conjoined representation.) Representationalism, 20, 35 seq.; 007 Principle, 20; change blindness, 35; computational/evolutionary disadvantages of, 35; middle ground, 36; Russell’s logical types as representationalist framework, 34. (See also Anti-representationalism.) Resemblance, see Signification. Residential recency, 86. Restricted numbers, 66 seq.; association with isolated, small-scale societies, 67, 129; assumed in Mesopotamia, 66 seq., 113 seq.; defined, 66 seq.; memory strategies, 76; Sumerian ternal counting, 118, 132. Restricted ordinal numbers, in Mesopotamian languages, refuted, 105. Retention of older forms for counting, 213; concurrent use with newer

forms, 213; patchwork of material forms, 213. (See also Distribution; Independence.) Roman numerals, see Latin numbers. Round Valley numbers, 73 seq. Rule sets, in numbers and language, 48. Russell, Bertrand, 225; cardinality, 32; failure of foundationalist enterprise, 29; importance of magnitude ordering, 25; logical types, 32 seq.; logical types as framework for categorizing in numbers, 33, 55; numbers as the recognition of cardinality shared between sets of objects, 3; typology, analytic usefulness of, 33. Script, cuneiform, 112 seq., 187 seq.; cursive, 193 seq.; Elamite, 22; Linear A, 22; types, 190. Seals, 163 seq., 187, 223. Semasiographic notations, 21 seq., 105 seq., 112; change in form of written numbers, 205 seq.; development of operational notations, 199; numbers, 196. (See also Writing, glottographic.) Semitic numbers, 120 seq. Sensory cortex, 61; topographic organization of, 17 seq., 60 seq. Set theory, 33. Sexagesimal, 115, 232; in Papua New Guinea, 233. (See also Place value.) Shona numbers, 77 seq. Sign, chronologies, 188; numerical systems, 182. Signification, 22, 106, 110 seq., 180, 185 seq., 224, 230; conventions and cultural knowledge, 186; representational modes: 187. (See also Ambiguity; Enactive signification; Icon; Index; Instantiation; Resemblance.) Simplicity, see Properties, writing. Siuai numbers, 77. Size effect, 44. (See also Distance effect; Number sense.) SNARC effect, 45. (See also Number sense.)

INDEX Societal inventiveness, pace of, 154. Socio-material complexity, correlation with numerical elaboration, 2, 19, 47, 88, 99, 128 seq., 217. (See also Material culture.) Somatic basis for numerical expression, 56, 71. Sora numbers, 84 seq., 138. South America, timeline for peopling, 46, 234. South American numbers, 70, 86. Spanish numbers, 22, 207 seq., 216, 220, 238. Spatial-numerical association of response codes, see SNARC effect. Speed of production, 193. (See also Script, cursive; Writing.) Spoken forms, recovery in ancient languages, 111. Stable order, 23; emergence of, 137; principle, in number systems, 137. (See also Ordinality.) Standardization, see Properties, writing. Stone tools, imposition of shape and symmetry, 2. Strategies employed by peoples with few numbers, 76. Structural persistence, see Material structure. Subitization, 16, 44; influence on language, 17, 93 seq., 100; influence on notational forms, 16 seq., 203 seq.; influence on numerical emergence, 70; influence on tokens, 175; measuring, 16; pairing, 34; psychological processes governing, 16, 44, 100, 103, 123; range, 15 seq.; symbolic and nonsymbolic quantities, 44. (See also Number sense.) Subtraction, 18, 25; in Oksapmin bodycounting, 27, 214; potential for devices to make explicit, 18; relations, 214. (See also Operations.) Successor function, 25, 242 seq. Sumerian numbers, 110, 132, 139, 216, 231, 234; Eblaite dialect, 120 seq.;

311 eme-sal, 110, 118 seq.; finger-counting in, 139; grammatical number, 113 seq., 121 seq.; lexical numbers, 113 seq., 139 seq.; ordinal numbers, 113 seq., 123, 126; productive grouping, 116; structure of, 115 seq.; ternal counting, 118, 131 seq.; younger than Akkadian, 116; writing reform, 198. Supramarginal gyrus, 58 seq. Synchronic view, limitations of, 92; numerical change, 69; numerical variability, 92. (See also Diachronic view.) Synchronization of material change and average traits, 194. Syntactic representation, 48. Syntax, 195. Tallies, 12, 26, 31, 40, 132, 134 seq., 175 seq., 198, 207 seq., 236; alternate uses, 148 seq.; challenges in archaeological deposition, preservation, and discovery, 146; cognitive opportunities, 145 seq., 149; compared to calculator, 12; criteria for discerning use in counting, 141 seq., 152; difficulty of establishing use in counting, 30; elaborational span, 140; ethnographic examples, 77, 79 seq.; evidence in the Ancient Near East, 4 seq., 146 seq., 151; exponential organization of, 202; geographic distribution of oldest sites with possible tallies, 148; grouping, 16; implicit relations, 145; implicit/explicit knowledge in, 201, 203, 207, 231, 237; implied use in early bookkeeping, 151; information and use, 145 seq.; invisibility in the archaeological record, 79; issues of continuity, 155; limitations of, 211; occasions for conceptual formation, 145; one-dimensional device, 65 seq., 78, 202; possible, from the Epipaleolithic Levant, 5; two-dimensional device, 82 seq., 220; transition between the body and material forms, 140, 153. (See also Affordances, tallies;

312

THE MATERIAL ORIGIN OF NUMBERS

Body-counting; Visual indistinguishability.) Tally, 1st millennium BCE, 152. Tally, 2nd millennium BCE, 151 seq. Ternal counting, see Sumerian numbers. Text corpora, 95; differences between modern and ancient languages, 104; realization in modern and ancient languages, 124; Sumerian, 124 seq. Thinking through vs. thinking about material forms, 241. Threshold of noticeability, see Magnitude appreciation; Perception. Tikopian numbers, 80. TM.75.G.02198, 3rd-millennium BCE lexical tablet, 115. Tokens, 1 seq., 157; archaeological, legal, and financial issues, 157, 164 seq.; assumptions about use in Mesopotamia, 3, 67, 113 seq., 163; association with conceptualization of numbers, 23; catalogue, 160, 166 seq., 168 seq., 254 seq.; complex, as precursor of writing, 160 seq., 164, 167, 174, 230; complexity, 163, 174; conjoined representation, 174, 184 seq., 223, 232; correspondences with impressions and numerical notations, 5, 157 seq., 162, 230 seq.; counting systems, 161 seq.; defined, 157; derived systems, 161; dimensionality, 202; discovery of numerical meaning, 1, 157; drawbacks, 177 seq.; exponential representation and implicit/explicit component, 233; geographic distribution of, 167, 170 seq.; identifying systems represented, 174; issues in interpreting as numerical counters, 164 seq.; likelihood of being numerical counters, 173; numerical meaning of, 157 seq., 161 seq., 166, 174, 180 seq., 210, 245 seq.; plain vs. complex, 160 seq.; plain, used in numeration, 161, 167, 172 seq.; precursor technologies, 154; relations, 173 seq.; sequence with later technologies, 159 seq., 164, 173; tem-

poral distribution of, 160, 163, 167, 170 seq., 173; two-dimensionality, 163; types, 167 seq.; unpublished finds, 166 seq. (See also Bundling/debundling; Grouping; Polyvalence; Relations; Visual indistinguishability; Ziyaret Tepe.) Tongan numbers, 68 seq., 82 seq., 208, 236. Tools, collaborative medium, 40 seq.; effects of use on groups, 24; synchronization of forms with average user capabilities, 24. Topological recognition of written objects, see Feature recognition. Touch, public, proscriptions against, 77, 83, 140. Transparency, etymological, material roots in language, 87; experiential, in material forms in cognition, 232, 238 seq. (See also Properties, material.) Trumaí numbers, 71. Tukano numbers, 72. Tylor, Edward Burnett, 3, 68. Typology, notational, Chrisomalis’ system, 202, 232 seq.; Zhang and Norman’s system, 232 seq. (See also Numerical notations.) UM 55-21-356, Old Babylonian mathematical tablet, 205. Unbundled maximum, see Bundling/debundling. Uruk periods as paleographic designations, 180. Variability, individual, cancelation through tool use by groups, 24. Visual appreciability, counting, 85 seq.; differences in numbers and writing, 197; numbers, alinguistic quality, 71 seq. Visual indistinguishability, cumulative notations, 203 seq.; fingers, 175; tallies, 144 seq., 175; tokens, 175. (See also Number sense; Properties, material.)

INDEX Visual Word Form Area, 189. (See also Feature recognition; Fusiform gyrus.) Visualization, 36, 38. W 20987,08, Pre-Uruk V clay bulla, 159. W 6066,a, Uruk IV administrative tablet, 185. W 6245,c, Uruk V administrative tablet, 205. Waimiri numbers, 94. Wappo numbers, 73 seq. Warao numbers, 83. Wathaurung numbers, 76 seq., 145. Weber–Fechner constant, 16, 44; grammatical number, 100; lexical numbers, 96; ordinal numbers, 103. WEIRD/non-WEIRD distinction, 51 seq.; grammatical number, 53; Müller-Lyer illusion, 51 seq.; quantity perception, 52, 66. (See also Grammatical number; Habituation; Number sense; Quantity appreciation.) Wernicke’s Area, 189. Women’s dialect, evolutionary development of, 119. (See also Sumerian numbers, eme-sal.) Word-frequency, see Frequency of use. Working memory, chimpanzees, 44; collaborative counting, 83; executive function, 228; freed to be repurposed, 217; limits in humans, 134; mental abacus, 62 seq.; process governing subitizing, 16, 100, 103, 122; use of material devices for counting, 85 seq.; visually complex characters, 189; writing, 217. (See also Attention; Automaticity; Habituation; Object tracking.)

313 Writing, ability to neurologically reorganize its users, 208; character recognition, 190; considered as material device, 5; development of literacy, 195; early interruption in Mesopotamia, 113 seq., 187 seq.; effects on numerical elaboration, 187, 197 seq.; expressiveness and fidelity to language, 111, 180, 195, 206; glottographic, 105 seq., 112 seq.; handwriting effects of, 192 seq., 197 seq., 204, 222 seq., 236; how writing means, 180; importance in mathematical elaboration, 199; invention of, 112, 197; language associated with earliest form in Mesopotamia, 113; lexical associations, 190 seq.; limited expressiveness in early forms, 113 seq.; material form, 179, 194; revolution, associated with expressiveness and literacy, 230; Sumerian ordinal frequency, 125 seq.; system for containing information, 229 seq.; timescale of change, 240; transforming relationship with information, 231; universality of process, 242. (See also Automaticity; Feature recognition; Habituation; Ideographs; Phonography; Pictographs; Properties, writing; Semasiographic notations.) Yanoama numbers, 71 seq. Yuki numbers, 73 seq., 87, 91, 96, 214. Yupno body-counting, 69; possible unreliability, 68. (See also Body-counting.) Ziyaret Tepe, 171; Archaeological Project, 167.