The Materiality of Numbers: Emergence and Elaboration from Prehistory to Present 1009361244, 9781009361248

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The Materiality of Numbers This is a book about numbers – what they are as concepts and how and why they originate – as viewed through the material devices used to represent and manipulate them. Fingers, tallies, tokens, and written notations, invented in both ancestral and contemporary societies, explain what numbers are, why they are the way they are, and how we get them. Cognitive archaeologist Karenleigh A. Overmann is the first to explore how material devices contribute to numerical thinking, initially by helping us to visualize and manipulate the perceptual experience of quantity that we share with other species. She explores how and why numbers are conceptualized and then elaborated, as well as the central role that material objects play in both processes. Overmann’s volume thus offers a view of numerical cognition that is based on an alternative set of assumptions about numbers, their material component, and the nature of the human mind and thinking.  .  earned her doctorate in archaeology from the University of Oxford as a Clarendon scholar after retiring from twenty-five years of active service in the US Navy. She currently directs the Center for Cognitive Archaeology at the University of Colorado, Colorado Springs.

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The Materiality of Numbers Emergence and Elaboration from Prehistory to Present  .  University of Colorado, Colorado Springs

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Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781009361248 DOI: 10.1017/9781009361262 © Karenleigh A. Overmann 2023 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press & Assessment. First published 2023 A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Overmann, Karenleigh A. (Karenleigh Anne), 1957- author. Title: The materiality of numbers : emergence and elaboration from prehistory to present / Karenleigh A. Overmann. Description: Cambridge ; New York, NY : Cambridge University Press, 2023. | Includes bibliographical references and index. Identifiers: LCCN 2022057067 (print) | LCCN 2022057068 (ebook) | ISBN 9781009361248 (hardback) | ISBN 9781009361255 (paperback) | ISBN 9781009361262 (epub) Subjects: LCSH: Numeration–History. Classification: LCC QA141.2 .O94 2023 (print) | LCC QA141.2 (ebook) | DDC 513.509–dc23/eng20230302 LC record available at https://lccn.loc.gov/2022057067 LC ebook record available at https://lccn.loc.gov/2022057068 ISBN 978-1-009-36124-8 Hardback Cambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

List of Figures List of Tables

page ix xii

Acknowledgments

xiii

Foreword

xiv

  Preface

1

xvii

Numbers in a Nutshell

1

What Numbers Are as Concepts

2

The Working Definition of Number

6

Analyzing Numbers through an Existing Familiarity

7

Who Has Numbers? And Do We All Have the Same Numbers?

9

Variability between Cultural Number Systems

13

Explaining Cross-Cultural Variability

16

Change within Any Particular Cultural Tradition

17

Are Numbers Ever the Same, and If So, How Are They the Same?

2

3

21

Converging Perspectives on Numbers

23

Historical Ideas about Where Numbers Come From

25

Four Recent Models of Numerical Origins

32

Historical Change in How We Study Number Systems

34

The Contributions and Challenges of Archaeology

41

The Brain in Numbers

43

Numerosity, the Innate Sense of Quantity

45

Categorization and Abstraction

51

v

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vi 

4

The Mental Number Line

58

The Parietal Lobe and Numbers

59

The Cerebellum and Numbers

64

Finger-Counting and the Brain

66

Bodies and Behaviors

70

How Numbers Emerge and the Roles of the Fingers

71

How Using the Fingers Patterns Numerical Structure and Language

5

6

77

Finger-Counting in Proto-Languages

82

The Visual Experience of the Hand

85

Movement and Material Engagement

89

Counting Behaviors When Numbers Are Few

91

Language in Numbers

94

What Language Reveals about Numbers

96

Lexical Numbers, Finger-Counting, and Analyzability

100

Evidence for Numerosity in Ancient Languages

103

Numerical Organization and Structure

112

Necessary and Sufficient Conditions

116

Global and Regional Patterns

125

The Global Pattern: Numerical Emergence and Prehistoric Migration

127

The Regional Pattern: Numerical Elaboration and SocioMaterial Complexity

7

130

Cultural Analogies

143

Materiality in Numbers

149

Recruiting New Devices as the Mechanism of Numerical Elaboration

150

Representing and Manipulating

163

Materially Anchored Conceptual Blending

164

General Effects of Material Culture

169

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 vii

8

9

Accumulating and Distributing Cognitive Effort

172

Change under Conditions of Common Use

174

Materiality in Cognition

176

Cognition Is Extended and Enactive

179

Materiality Has Agency

185

Enactive Signification

190

Sustained Collaboration, Common Creativity

192

Making Quantity Tangible and Manipulable

195

Gaining Control of Quantity Percepts

198

The Instrumental Body

202

Using the Hand as a Material Device for Counting

206

The Hand’s Influence on Verbal Expressions

212

Finger-Counting Age

217

10 Tallies and Other Devices That Accumulate

220

Devices That Accumulate

224

Tallying and Numerical Organization and Structure

226

Taking Up the Material (Noncorporeal) Tally

228

Tokenization

234

Tallying without Words

239

11 Interpreting Prehistoric Artifacts

244

Mark-Making and Prehistoric Numbers

247

Archaeological Techniques for Interpreting Prehistoric Marks

254

A Cautionary Tale: The Australian Message Sticks

260

Interpreting Palaeolithic Artifacts as Possible Tallies

263

Functional Considerations

267

12 Devices That Accumulate and Group

277

The Inka Counting Board

280

Oceanian Counting by Sorting and the “Ephemeral Abacus”

288

The Yoruba Cowrie Abacus

298

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viii 

13 Handwritten Notations

309

Contiguity of Function between Numerical Notations and Precursor Technologies

312

Differences between Written Numbers and Writing for Nonnumerical Language

320

What Writing – Numerical and Not – Adds to Numbers

332

Major Trends

336

14 The Materiality of Numbers

340

Future Directions in Research

349

A Final Thought

351

References

354

Index

406

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Figures

1.1 The working definition of number 1.2 Possible prehistoric counting devices

page 6 8

1.3 The mental number line

15

1.4 Woodcut from Margarita Philosophica

20

2.1 The Müller-Lyer illusion

39

3.1 Subitization and magnitude appreciation

45

3.2 Frequency of use for the lexical numbers one through twenty

48

3.3 Homo sapiens and Neandertal skulls

60

3.4 The cortical homunculus

68

3.5 Topographical layout of the motor and somatosensory cortices 4.1 Oksapmin body-counting

69 81

5.1 Frequency of use for the ordinal numbers first through eighth

105

5.2 Sumerian ordinal frequency for the ordinals first through twentieth

107

5.3 Organization and structure of the Sumerian lexical numbers

113

6.1 Geographic distribution of analyzable number-words

128

6.2 The Upper Rio Negro cultural area

133

7.1 Administrative tablet (Sb 22218) from Susa, modern Iran

161

7.2 Chief accountant holding a khipu

165

7.3 Materially anchored conceptual blending

168

9.1 Visual choices

201

9.2 Using the hand and fingers in restricted counting

207

9.3 Using more positions on the same body

207

9.4 Reusing the fingers of the same person

209 ix

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x    9.5 Recruiting additional bodies

210

9.6 Two hand stencils from Cosquer Cave

218

10.1 Pieces of incised wood made by Baniwa and Yuruti men 11.1 Engraved baboon fibula from Border Cave, South Africa

231 245

11.2 Three Australian message sticks from the nineteenth century 11.3 Engraved aurochs bone from Nesher Ramla, Israel

262 268

11.4 Engraved hyena femur from Mousterian site at Les Pradelles, France

271

11.5 Two views of an engraved bovine rib fragment from the Grotte du Taï, France

273

11.6 Engraved bone from Ishango, Democratic Republic of the Congo

275

12.1 Devices that accumulate and group

278

12.2 Khipu and yupana

280

12.3 Four interpretations of Guamán Poma’s counting board

285

12.4 Positional and additive numerals and corresponding representations in khipu knots

287

12.5 Polynesian tally-counting

291

12.6 Mangarevan binary counting

294

12.7 Yoruba numbers and the cowrie bundles they are based on 12.8 Twenty-by-twenty cowrie grid

299 305

13.1 Instantiation: to represent by being an instance of something

310

13.2 Correspondences of shape and size (tokens, impressions, notations)

313

13.3 Inka khipu made of knotted cotton or wool

317

13.4 Notional Inka khipu

318

13.5 Signification: to represent by depicting, indicating, or suggesting something

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322

   xi 13.6 Change in written numerals over five millennia and multiple languages 13.7 Loss of depictiveness in early writing

323 325

13.8 Administrative tablet (TM.75.G.2198) from Ebla, modern Syria

329

13.9 Table of reciprocals of 60 (ERM 14645), provenience unknown

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334

Tables

3.1 Categorical judgments

page 52

3.2 Russell’s logical types

55

4.1 Oksapmin number-words

82

4.2 Proto-Indo-European numbers

83

4.3 Desana numbers

84

5.1 Sumerian and Akkadian words for the numbers one through ten

101

5.2 Grammatical number in Sumerian, Akkadian, Elamite, and English

110

5.3 Chinese and English written numerals and spoken numbers

117

6.1 Regional comparison of transitions in subsistence, sedentism, and population

130

7.1 Types and chronology of material devices used in numbers

152

11.1 Prehistoric artifacts with possible numerical intent, purpose, or meaning 12.1 Ten ways to form the number 19,669 in Yoruba

264 302

12.2 Possible variations of movements on the Yoruba cowrie grid

307

13.1 Sumerian nonphonetic and Akkadian phonetic notations

xii

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331

Acknowledgments

This book was made possible only through the encouragement and expertise of many individuals, to whom I owe enormous debts of gratitude, both personal and intellectual. First and always is my husband, Bill, my best friend and an unfailing source of support, and of course my children, Will (1987–2016), Archie, and Barbara and their wonderful families. Next are Tom Wynn and Brian Hayden, whose detailed and challenging critiques greatly improved the manuscript, and Lambros Malafouris and Rex Welshon, who helped me appreciate the role of materiality in human cognition. Fred Coolidge encouraged me to take up graduate work when my inclination had been to add a fourth undergraduate major, Joan Ray guided my entry into academic publishing, and Colin Renfrew enjoined me to do something new. I also want to thank my Assyriology colleagues who have helped and inspired me along the way, particularly Jacob Dahl, Bob Englund, Jens Høyrup, Annick Payne, Denise Schmandt-Besserat, and Klaus Wagensonner; the many colleagues in archaeology, numerical cognition, and writing systems research whose work has influenced and challenged me, including Sieghard Beller, Andrea Bender, Thiago Chacon, Steve Chrisomalis, Marie Coppola, Virginie Crollen, Francesco d’Errico, Pattie Epps, Silvia Ferrara, Cinzia Florio, Chris Gosden, Piers Kelly, Rafael Núñez, Pierre Pica, Dwight Read, André Rouillon, Geoff Saxe, Oliver Schlaudt, Dirk Schlimm, Gary Urton, and Larry Vandevert; my friends and colleagues at the University of Bergen, especially Vilde Aarethun, Gisela Böhm, Hege Bye, Samantha Harris, Marina Hirnstein, Dixie Brea Larios, Tony Leino, Torill Christine Lindstrøm, Valeria Markova, Edwin Nordstrom, and Larissa Mendoza Straffon; and Rudie Botha and an anonymous reader for Cambridge University Press for their generous assessment of the manuscript. xiii

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Foreword

If, like me, you grew up in the industrialized world, you have been surrounded by numbers almost from birth. Our modern world is immersed in numbers. We encounter them everywhere – on money, calendars, highway signs, airline flights, even our official identities. Numbers lie at the core of mathematics, where they also reveal complex relationships that have enabled our ever more complex hegemony over the natural world. Ability to handle numbers is rivaled only by ability in language when we assess human intelligence. And like language, there are regions of our brains that appear to be dedicated to numerical and mathematical understanding. Numbers are a way to think about quantity. However, they are not natural phenomena. There are no numbers in the natural world waiting to be discovered. The natural world does have ways to think about quantity that evolved long ago and are shared by many different animal species. One is the ability to distinguish sets of , and sometimes

,

,

. The other is the ability to

compare sets of items to one another and judge which is larger. But these are not numbers. Numbers are ordinal – they extend in an invariant sequence – and they have cardinal value; each successive number advances in value by the same amount (1, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, etc.). But how did we acquire this unnatural but very useful way to think about quantity? This is the question that Karenleigh Overmann tackles in this book, and it is a fascinating account. Dr. Overmann examines an array of evidence, including archaeology (from the Stone Age through the rise of civilization), brain anatomy and function, nonhuman primates, languages (both living and dead), and the geographic distribution of counting systems. But the weight of her reasoning rests on the comparative ethnographic evidence of human counting from around xiv

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 xv the world and from a variety of technical and cultural contexts. Not all humans have numbers, or even count. Some societies have words for only one, two, and three; any quantity greater is “many.” Others have systems of counting that match items with digits (fingers and toes), and occasionally other body parts – a hand’s worth of shells or an elbow’s worth of yams. And yet others use tokens, usually small manipulable objects, to expand the range of counting possible with body parts. Dr. Overmann marshals this evidence brilliantly into an account of how numbers must have emerged in the prehistoric and early historic past. But unlike many other scholars who have examined this evidence, Dr. Overmann does not consider these fingers and tokens to be expressions of a number system; instead, she presents them as the very resources through and by which people created numbers in the first place. Think for a moment about how children learn numbers. Most often they first learn a list of number labels, sometimes extending well past twenty. But they do not yet understand the cardinal nature of numbers – the rule that generates a sequence running to infinity. They then use their memorized list as an aid to build an understanding of numbers – first the smaller values (3 is one more than 2), then larger values (14 is one more than 13), until ultimately, they acquire an understanding of the rule itself. Cognitive science labels such aids “scaffolds.” But what about people who possessed no such scaffold, that is, people who had no labels for numbers? How could they reach an understanding of cardinality and ordinality? They couldn’t have “discovered” them because nature does not provide any cardinal or even ordinal sequences. But they did have potential aids in the form of fingers and toes and material objects. Dr. Overmann documents clearly how such manipulation, over hundreds of generations, yielded our understanding of number. Our ancestors invented numbers, and they did it by manipulating material things. Dr. Overmann’s achievement is arguably the best explicit example of evolutionary extended cognition yet proposed. Models of

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xvi  embodied and extended cognition have been very much in vogue in some academic circles for over thirty years now. The idea that the mind is made up not of just gray (really, pink) matter in the head, but also includes bodily resources and cultural extensions is popular because it sidesteps epistemological problems attendant with the classic Cartesian dualism between minds and bodies. However, most of the literature remains primarily philosophical and theoretical, with the occasional thought experiment in support (e.g., Merleau-Ponty’s blind man’s stick). The approach has made few inroads into the empirical science of human evolution. Dr. Overmann’s study of number is thus badly needed, providing a persuasive account of the active role played by materiality in the recent evolution of the human mind. Dr. Overmann’s analysis has significance beyond the narrow domain of numerical cognition. By documenting the active role of materiality in the development of numerical thinking, she opens the door for consideration of materiality as a component of other cognitive systems. Cognitive archaeology – my particular field of expertise – is a recent addition to the corpus of methods that science deploys to investigate the evolution of mind, and it has always been hamstrung by the Cartesian mind-body dualism of traditional cognitive science. Archaeology studies material things and must infer the components of mindedness. If material things played an active role in cognitive evolution, then archaeologists’ burden of proof would be lightened considerably. But it would still require close reasoning and careful marshaling of evidence from a variety of domains. Dr. Overmann shows the way in this remarkable volume. Thomas Wynn Distinguished Professor of Anthropology Emeritus, University of Colorado, Colorado Springs September 2022

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Preface

Early in the sixteenth century, Spanish conquistadors first encountered the khipu, the device of knotted strings used by the Inka to record numbers and other information.1 The khipu astonished the Spaniards, who clearly did not understand it as a form of writing – not even its numerical component, which shared many of its qualities with the various forms of abacus prevalent in Europe at the time.2 Since the sixteenth century, colonial chroniclers and modern authors alike have spoken of the khipu as a device for computation. However, khipus could not have been used for this purpose, since knots cannot possibly be tied, untied, and retied quickly enough for the effective manipulation that computation requires. This is one of the main differences that the khipu has with an abacus, which has the manipulability needed for calculating because its beads are separate, and thus rearrangeable, objects. What follows from this persistent misunderstanding of the khipu is this: We generally lack insight into the material devices used for numbers, what they can and cannot do in representing and manipulating numbers, and why this is important to understanding what numbers are and where they come from. This book attempts an answer. Certainly, a lot of books have already been written about numbers, including their origin, and from a wide variety of perspectives: psychological, developmental, neuroscientific, comparative, linguistic, cultural/ethnographic, historical, philosophical, and of course, mathematical. This particular book differs because it addresses something that all the others mention

1

2

De Acosta, 1590; de la Vega, 1609; Guamán Poma, 1615. Khipu is the Quechan word for “knot”; the word is spelled quipu or quipo in Spanish. Pullan, 1968.

xvii

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xviii  only in passing, if at all: the material devices used to represent and manipulate numerical concepts, things like fingers and tallies and tokens and notations. These devices are important because they can explain what numbers are and why they are the way they are. They are the material aspect or materiality of numbers; they make numbers material and thus tangible and rearrangeable as forms that occasion the emergence of new numerical properties and meanings.3 There are three reasons why the materiality of numbers is so critical and yet so frequently invisible or ignored. First, our perceptual system for quantity has a great deal to do with numbers, since it patterns their initial expression and informs their content, visualization, and manipulation as numbers become more elaborate. This perceptual system is known as numerosity or the number sense.4 The latter term is misleading because what we perceive is quantity, not number, a nontrivial distinction between experience and conceptualization.5 While numerosity is multimodal – that is, we can perceive quantity not just visually but also through hearing6 and touch7 – only the visual dimension permits our quantity percepts to be manipulated into new, concept-generating stimuli that can be attended simultaneously. Simply, our auditory and tactile senses lack the width and granularity of focus needed to attend to multiple, parallel instances of number. Not only can material forms enable us to do this, but they also let us take control of the quantity that we visually perceive and enable us to rearrange it until it makes different sense to us. This is one of the reasons that the material forms we use to represent and manipulate numbers are critically important to their realization and elaboration as concepts. The second reason why the material aspect of numbers is critical but neglected is that the numbers we have today, the numbers

3

4 6

I am indebted to Lambros Malafouris for the idea that materiality makes the ineffable sense of quantity tangible and manipulable. This view was foundational to the argument presented here of how numbers are realized and elaborated. 5 Dehaene, 2011. Núñez, 2017a, 2017b. 7 Jordan et al., 2005, 2008; Beran, 2012. Davis et al., 1989; Krause et al., 2013.

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 xix of the Western tradition, are thousands of years old – likely tens of thousands of years old, and perhaps even 100,000 years old or more. During this impressive cultural persistence, they have not only acquired properties from the material devices used to represent and manipulate them, but they have also become distributed over multiple, differing devices – fingers, tallies, tokens, and numerical notations – along with all the things that display numerical representations, from clocks to coins, and the things to which they are applied, from angels to zebras. This distribution has made numbers functionally independent of any particular device; independence, in turn, makes numbers appear to be an in-the-head phenomenon, something that does not require any material devices.8 In fact, the idea that numbers are a mental phenomenon is a fundamental assumption that pervades the psychological, developmental, neuroscientific, comparative, linguistic, cultural/ethnographic, historical, philosophical, and mathematical literature on numbers. A related and quite common assumption is the idea that the material devices used for numbers are mere passive repositories for that mental content: The brain thinks something up and then externalizes it onto a material form. In this book, the opposing view is taken: Material devices are not passive repositories, but rather, active participants in the cognitive system for numbers. They are active because they influence and systematize how numbers emerge and become elaborated. A third reason the material aspect is neglected, one that is not inconsiderable, is that recognizing and then putting the materiality back into numbers requires stretching the definition of what constitutes a material form beyond an easy recognizability. For example, the fingers used in counting can be considered a material device, and so can the temporary structures created by the various sorting and collaborative strategies found in Oceania and Africa, and so too can symbolic notations like the Western numbers 0 through 9. 8

Overmann, 2019a, 2019b.

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xx  As material forms, such things obviously differ in the nature of their materiality from the actual artifacts used in counting, things like tallies, tokens, counting boards, and calculators. In these unconventional cases, the structures in question can be recognized both as material and as possessing unique attributes. The fingers are material but also composed of living flesh; sorting and collaborative strategies produce temporary material forms whose ephemerality differs from the solidity and persistence of devices like the abacus; and symbolic notations, while material, possess an arbitrary, conventional form. Arguably, these other attributes do not necessitate the exclusion of such forms from being considered for their material qualities. In fact, considering unconventional devices for their material quality is analytically useful. It lets us see how material forms act to structure numbers and infuse them with properties like linearity, stable order, capacity, manipulability, and conciseness. Importantly, it allows us to reconstruct and thus connect the entire span of numerical elaboration, from the perception of quantity shared with other species to symbolic notations. What connects numerosity to notations and the associated concepts of numbers is the manuovisual manipulation of material forms, a behavior exhibited by the human species alone. The initial premise of the argument presented here is straightforward. Today, symbolic notations are understood as having a role in how we acquire numerical concepts and in what we understand them to be.9 Symbolic notations – especially early forms like five as represented by vertical wedges, as in Babylonian cuneiform – share affinities of form and function with the technologies that preceded them, things like fingers (vertically extended digits) and tally marks (vertical parallel lines). Accordingly, we can ascribe the same roles that notations have in conceptual acquisition and content to their precursors, and these precursors systematically connect symbolic notations to the initial perception of 9

Schlimm, 2018.

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 xxi quantity. This is not surprising, since after all, before and without symbolic notations, societies both ancient and contemporary have used devices like fingers, tallies, and abaci to record numbers and calculate with them, and quite capably too. Numbers have been used and taught for thousands of years, and for most of that history and prehistory, they were used and taught without symbolic notations, a form that is relatively recent in the chronology.10 From this starting point, this book explains the role of material forms within the cognitive system for numbers, focusing on how numbers emerge and become elaborated. For example, new material forms are systematically incorporated to address limitations of previous forms in a way that is highly similar across cultures; new material forms add new properties to numbers to act thereby as the mechanism of numerical elaboration; and as new forms are added, the explicit representational form becomes more concise, while an implicit component, the knowledge that the user must supply, concomitantly increases.11 Seeing numbers as a system realized and elaborated through the use of material devices can require more than an unconventional view of what constitutes a material device for numbers: It may also require an unconventional view of cognition itself. In this book, cognition is considered to be a system that consists of bodies and material forms, in addition to the brain. That is, the mind is seen as embodied, embedded, extended, and enactive (4E). The theoretical framework used to analyze cognition as a 4E system distributed over time is the Material Engagement Theory (MET) of archaeologist Lambros Malafouris.12 A 4E framework like MET is vital to the field of evolutionary cognitive archaeology, the discipline that seeks to understand how the human mind evolved by examining the material record of ancient societies and extinct hominin species. Treating the mind as 4E opens up a dimension of temporal change that cannot otherwise be easily investigated. The MET framework elevates the 10

Adkins, 1956, p. 1.

11

Overmann, 2019b.

12

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Malafouris, 2013.

xxii  material record from passive traces of behaviors to actual components of ancient minds.13 This in turn permits the examination of a temporal dimension of change over time that disciplines like contemporary neuroscience cannot match in either depth or granularity. Focusing on the materiality of numbers does not mean that the other domains investigating numbers are unimportant. Far from it, for they are often crucial to understanding not just numbers but also how material forms function in numbers. For example, the neurological interconnections and interactions between the different parts of the brain involved in numbers explain why finger-counting, the use of the hand as a material device for the representation and some manipulation of numbers, is so cross-culturally ubiquitous. Language, considered here to be another means of accessing and expressing our numerical intuitions, is often a source of evidence that the fingers were once used for counting, not just in the global preponderance of ten, five, and twenty as the organizing basis for numbers, but also in formations like six that mean five plus one and imply that the fingers were used to explicate and then label numerical concepts. Conversely, the hand being used as a material device explains why its patterns of use and associated linguistic labeling differ cross-culturally, since despite the fact that all humans have essentially the same hand and behavior in using it for counting, societies use the hand differently in counting. In other words, despite the common assumption that numbers are purely mental constructs, there does not seem to be any internal (that is, mental or innate) mandate for using the hand for counting in any specific way. This leaves different societies free to make unique choices in how the hand is used, creating a wide and occasionally astonishing cross-cultural variability. It will also be useful to explore some of the assumptions made by the other disciplines that investigate numeracy, the ability to reason with numbers that presumes numerical concepts are available. 13

Wynn et al., 2021.

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 xxiii For example, psychologists looking at numeracy in humans and cognitive scientists focusing on numerical abilities in animals often see numerosity as all that is needed for conceptualizing numerical concepts and performing simple computations. Accordingly, they find numerical concepts and operations in species as divergent as chimpanzees and honeybees.14 This is the nativist view that mental capacities and structures are innate or biologically endowed, rather than learned. Be that as it may, if the nativist view were correct, then linguists, who argue that language is the reason we have number concepts, cannot be right, since only humans have language.15 The exact opposite is true as well, since if language were the source of numbers and numeracy, then as only humans have language, only humans would have numbers. In fact, both positions are problematic because all humans have numerosity and language, but not all human societies have numbers, at least not to the degree found in the Western numerical tradition. Some societies, like the Mundurukú of Amazonian Brazil, have very few numbers16 – one, two, about three, and about four – and one of their neighbors, the Pirahã, are said to have no numbers whatsoever.17 If reports of Mundurukú and Pirahã anumeracy are more severe than their actuality, the degree to which some societies do lack numbers is nonetheless a severe challenge to the positions that numerosity and/or language comprise all that are/is needed for numbers.18 Pace those holding such positions, these shortfalls create explanatory gaps that material forms have the potential to answer. Accordingly, it is argued here that material forms are also needed: not just material culture in general, but specific material devices for representing and manipulating numbers – however unconventionally defined – as motivated by specific socio-material conditions. Along the way, many fascinating number systems and devices will be

14 15 17

Nieder, 2017a, 2017b; Howard et al., 2018; Clarke & Beck, 2021. 16 Overmann, 2021e. Pica & Lecomte, 2008; Rooryck et al., 2017. 18 Everett, 2005, 2007, 2013; Frank et al., 2008. Overmann, 2021e.

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xxiv  introduced. We will also see how and why devices work, what they can and cannot do as material forms, and why this too is important. The book proceeds as follows:  Chapters 1 and 2 look at what numbers are as concepts and historical and contemporary ideas about their origin.  Chapters 3 through 5 examine some of the major things contributing to what numbers are – perceptual functions like numerosity, behavioral strategies like finger-counting and pairing, and language – while focusing on how these domains contribute to our understanding of the material component.  Chapter 6 examines how number systems are distributed globally and regionally; these nonrandom distributions let us see number systems in Australia and South America as presently emerging, providing critical insight into a panhuman process.  Chapter 7 introduces key ideas about materiality and delineates the process whereby the incorporation and use of material forms systematically produces and elaborates numerical concepts.  Chapter 8 reviews the theoretical framework used, Material Engagement Theory, and shows why the extended/enactive model of cognition is both useful and valid.  Chapter 9 examines how quantity percepts are made tangible and manipulable through the use of distributed exemplars (these are features of the environment whose reliable quantity is recruited to express that number), the fingers, and small, manipulable objects.  Chapter 10 deals with material forms like the tally that accumulate, and examines the implications of tokenization, the idea that deindividuated elements are important to numbers and the process whereby elements like notches might become deindividuated.  Chapter 11 introduces the archaeological techniques currently being used to determine whether prehistoric artifacts were tallies, along with the shortfalls of such techniques.  Chapter 12 is concerned with highly elaborated number systems that use material devices that accumulate and group, but not symbolic notations; this class of number systems is often not well understood.  Chapter 13 examines numerical notations, what they add to numerical elaboration, how they differ from other written signs, and why their qualities are important.

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 xxv  Finally, Chapter 14 sums things up by looking at the way ahead for this fascinating, promising, and unique research into numbers.

The view that numbers are essentially material in their origin and elaboration is a novel perspective, one that the philosopher Karim Zahidi recently called a “naturalistically plausible account of the emergence of the modern natural number concept.”19 This perspective is new. It has not been tried in at least the 2500 years since Plato pronounced that numbers were mental constructs – eternal but intangible and invisible concepts existing somewhere external to the human minds that somehow still managed to contact and apprehend them. In explaining this new perspective on numerical origin and elaboration, this book draws its conclusions from a different set of assumptions about numbers, their material component, and the nature of the human mind.

19

Zahidi, 2021, p. 530.

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

Numbers in a Nutshell

This is a book about what numbers are and where they come from, as understood through their materiality, the material devices used to represent and manipulate them: things like fingers, tallies, tokens, and symbolic notations. This book is concerned with the natural or counting numbers – the sequence one, two, three, four, and so on, and maybe as high as ten or twenty or hundred – that are the basis of arithmetic and mathematics. While the book focuses on how concepts of number emerge and ultimately become elaborated as arithmetic and mathematics through the use of material devices, it will also examine related phenomena, like the way numbers vary crossculturally. This book examines numbers through the lens of archaeology. Why archaeology, of all things, is a reasonable question, since numbers are not the sort of thing that can be dug up from the ground or analyzed in the lab, the activities typically performed by most archaeologists. However, archaeology is also the science of material objects, and here we are looking at numbers through their material component, the counting devices used to represent and manipulate them. These devices include distributed exemplars (these are objects like the arms or the hand, whose dependable quantity is used to express quantities like two and five); the fingers used in counting; tallies and other devices that accumulate quantity; tokens and forms like the abacus that accumulate, group, and permit the manipulation of quantity; and numerical notations. As noted in the preface, some of these forms are unconventional as material devices, but will be treated as such for the purposes of this analysis. We are also taking a cognitive approach to material objects. Accordingly, we will consider how and why material objects 

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     contribute to numerical concepts and numerical thinking, past and present. This will require us to consider the material devices used in numbers as having a role in conceptualizing and thinking about numbers. We will consider material devices to be an implicit part of the cognitive system for numbers, and this approach and the theoretical framework used are explained in later chapters. To understand what material devices do in numerical conceptualization and thinking, we will also need to look beyond the archaeological data and consider data from other disciplines, particularly psychology and neuroscience, paleoneurology, biological anthropology and zoology, linguistics, and ethnography. The interdisciplinary data provide information that is useful for attesting or explaining how material forms function in numbers. For example, contemporary languages often attest to ancient finger-counting in forms like six that mean five and one and in productive terms that show counting structured by the number of fingers, like ten (the number of the fingers) and hundred (the number of the fingers counted by the same amount). Similarly, neuropsychology provides insight into neural interconnections within the brain that explain why finger-counting is ubiquitous and cross-culturally prevalent. Such data are also useful for understanding what numbers are as concepts. This understanding is vital when investigating the questions of how, when, and why numbers began, as it necessarily informs what we look for in the archaeological record and how we interpret what we find there. Thus, we will begin by looking at what numbers are as concepts.

     Number is formally defined as “a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication; an element (such as π) of any of many mathematical systems obtained by extension of or analogy with the natural number system.” (Merriam-Webster, 2014, def. 1c2 and 1c3)

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     As formally if somewhat circularly defined, a number is an element of a mathematical system obtained by extending or analogizing the natural numbers,1 which are also known as the counting numbers, the whole numbers, or the integers – one, two, three, and so on. Numbers are the basic elements of a mathematical system, so all of the things that we think of as arithmetic and mathematics develop – or have the potential to develop – once a basic counting sequence is available. As stated, the formal definition is arguably an unsatisfactory basis for our stated goal, which is understanding numerical emergence and elaboration through the material devices used for representing and manipulating numbers. We need a definition that specifies numbers in terms of their properties – particularly those properties that can be associated with and explained by the material devices used, and which can be empirically established through the devices and properties of different cultural number systems. We will start by considering the old and deeply philosophical questions of what numbers are as concepts – what the Greek philosopher Aristotle might have called their essence, the properties that give an entity or a substance its identity and nature. Here we will examine what numbers are as concepts by specifying their properties. A number, first and foremost, is the idea of how many of something there are, a distinct or discrete amount. This is cardinality, or how many of something there are in a group of objects. For example, a trio has three members, a property of threeness, and the number three is how many members all trios have in common.2 In offering this definition, the philosopher and mathematician Bertrand Russell distinguished a property of a particular trio (threeness) from a property shared and instantiated by all trios (the number three). The former is the property of having three members and is applicable to a particular trio. The latter is a number, a property of all the sets with that many members. The distinction between the quantity of a particular set and 1

Merriam-Webster, 2014.

2

Russell, 1920.

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     the idea that a number is a quantity shared by two or more sets is consistent with the idea that number begins as the perception of quantity: The first is something we can appreciate through the perceptual system for quantity when there are no more than three or four members, while the second is the conceptualization of that quantity as a number. We will look at how material forms are used as a second (or “reference”) set to express perceptible quantity, which helps us visualize, understand, and express quantities that lie beyond the perceptible range of about three or four. A number also has a specific place in a counting sequence. This is ordinality, numbers in order. For example, six is the number between five and seven. In any counting sequence, numerical order is based on increasing size: It is five, six, seven, eight, and never six, five, eight, seven or any of the other permutations possible – though granted, the sequence eight, seven, six, five might preface an annual cheering of Happy New Year! in Times Square or follow the phrase “ten seconds to liftoff” at NASA. When whole numbers or integers are counted in sequence, each number is one more than the one it follows. In the sequence one, two, three, three follows two and is one more than two, and two follows one and is one more than one. While the relation of one-more is implicit to an ordinal sequence of counting numbers, it is not necessarily explicit. After all, ordinality is no more than ordering, and as such, is as equally applicable to sequences like the letters of the alphabet or the days of the week as it is to a sequence of counting numbers. Ordinality does not fix the interval between any of the members of any sequence. Discovering that the interval between counting numbers is one is a matter of using material devices, where each new notch on a tally, for example, can be visually discerned as one-more than the previous notch in the process of making them. Numbers have the potential for many more relations between them than just one-more. For example, six is the result of adding four and two, one of the many additive combinations that produce this number; others are three plus three, five plus one, eight minus two,

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     and thirteen minus seven. Even one, two, and three are potentially related to each other in more ways than just the one-more of an ordinal counting sequence, since for example, three is two more than one. Just like the explicit one-more relation between sequential numbers was a matter of elaboration, so too are any other explicit relations between numbers. What is required for such elaboration is a manipulable technology like pebbles or tokens, objects that can be rearranged into different subgroups. Numbers – or rather, the relations between them – have the potential to be manipulated by means of operations like addition and subtraction. Operations can involve explicit relations between numbers. For example, knowing the relations between two, four, and six permits the addition of two and four to obtain six, the subtraction of two from six to obtain four, and the subtraction of four from six to obtain two. It is also possible to add and subtract without explicit relations. For example, two groups of like objects can be commingled, and the whole counted to obtain the total without knowing any relations between numbers. This is true of numerical counters as well, since the beads on an abacus can be moved without the numerical relations being explicit. In any case, when relations are explicit, they facilitate the ability to compute mentally, rather than mechanically. Such relations are essential to mental – or, more accurately, knowledge-based – calculation. The corollary to that thought is this: When such relations do not yet exist, knowledge-based calculation is not yet possible. We will look at how material forms support the emergence of mechanical and knowledge-based calculation. Not all numbers have attributes like the meshwork of potential relations – for example, two being the square root of four and the difference between 1,245,762 and both 1,245,760 and 1,245,764 – that characterize Western numbers. These are numbers in a decimal or base 10 system typically written in the familiar Hindu–Arabic notations (0 through 9). We are particularly interested in the differences between cultural number systems, not only because they are

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     fascinating, but also because they are potential clues to where numbers come from and how they become elaborated over time.

     The working definition of number used here is this: Numbers are concepts of discrete quantity, arranged in magnitude order, with relations between them, and operations that manipulate the relations (Fig. 1.1). As a system of numbers elaborates, it will also acquire a productive base, a number upon which other numbers are built. For example, in Western numbers, the number ten serves as the

. . The working definition of number. The definition focuses on five key properties: discreteness, magnitude ordering, relatedness, operational manipulability, and productive grouping. Image by the author.

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     productive base, as it is repeatedly added or multiplied to produce values like twenty (either 10 + 10 or 10  2), thirty (10 + 10 + 10 or 10  3), hundred (ten tens or 10  10), and thousand (ten hundreds or 10  10  10). These qualities are simply, no more and no less, what Western readers will already know about numbers from what they have been exposed to through culture and language and have learned through formal education. Granted, many readers may not have thought explicitly about numbers in terms of such properties before. Readers have also learned algorithms, or sequences of operations, that enable them to do things like add columns of numbers, divide one number into another regardless of which one is larger, and convert fractions from ratios to decimal format. While algorithmic insight will not be much called upon here – since our interest lies more in how such computations are performed, rather than performing such computations – readers can nonetheless use their existing knowledge of numbers and computations as a basis for gaining new insights into how such things become elaborated from a sequence of counting numbers, say, the numbers one through ten.

      People are enculturated into the numbers of their society from day one. For example, people in the Western tradition are exposed to objects that have quantity and can be counted; social behaviors like counting and finger-counting; social purposes like inventorying that involve numbers; material representations of numbers like written symbols and tally marks; and different forms of numbers in language. This means that most readers will have a considerable knowledge of numbers, whether or not that knowledge is explicit in the particular ways used here. Something to keep in mind about our familiar Western numbers is that the Western numerical tradition is quite old. Its roots lie deep in the world’s ancient mathematical traditions, those of Rome,

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    

. . Possible prehistoric counting devices. (Top) Notched bone from Border Cave, South Africa, dated to approximately 42,000 years ago. (Bottom) Shells punched to be strung from Blombos Cave, South Africa, dated to approximately 77,000 years ago. Top image adapted from d’Errico et al. (2012, Supporting Information, Fig. 9, top image). Image © PNAS and used with permission. Bottom image adapted from one by Christopher Henshilwood and Francesco d’Errico, distributed under a Creative Commons license.

Greece, India, Egypt, and Babylon, traditions with even deeper temporal roots in counting sequences and practices that would have developed during the Neolithic and Upper Palaeolithic. The world’s earliest known unambiguous numbers are numerical impressions in clay found in Mesopotamia in the mid-fourth millennium BCE.3 Since Mesopotamian numbers are one of their roots, this makes Western numbers at least 5000 or 6000 years old. Undoubtedly, Western numbers are considerably older – perhaps 20,000 or 30,000 years old – given that the Mesopotamian numbers were already significantly elaborated by the time they first appear in the archaeological record. As if this timespan were not already impressive enough, Western numbers are likely to be older still, if archaeologists are correct in interpreting 42,000-year-old notched bones as tallies4 (Fig. 1.2, top) and 77,000-year-old shell beads as rosaries5 (Fig. 1.2, bottom). This impressive lifespan means that Western numbers have had a lot of time to change, and indeed, they have become highly elaborated, acquiring properties that are not necessarily shared by numbers in other cultural traditions. 3 4 5

Schmandt-Besserat, 1992a; Nissen et al., 1993; Overmann, 2016b, 2019b. Beaumont, 1973; d’Errico et al., 2012. Henshilwood et al., 2004; d’Errico et al., 2005.

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     Readers’ knowledge of the highly elaborated Western numerical concepts produced by this lengthy history and prehistory is a valuable resource for understanding the numbers of other cultural traditions. The key is thinking analytically about what is already known: This can help in understanding the ways in which other cultural number systems differ from the Western tradition, and in appreciating the principles of content, organization, and structure illuminated by the differences.

  ?        ? Most, but not all, human societies have numbers. And while all societies that have numbers develop ones that are highly similar in their content, structure, and organization, no two societies develop identical number systems. We will look at differences and similarities between numerical traditions and the reasons for these differences and similarities. A major reason for similarity is that numbers emerge from the same perceptual experience of quantity and are represented with the same devices, things like the hands. Another reason, one that complicates the attempt to understand numerical emergence and elaboration, is that societies often borrow the numbers developed by another. Today, many societies have adopted Western numbers, just as the West once adopted the Hindu–Arabic notations and used them alongside the Roman numerals that have since become an archaic system retained for its prestige value.6 The current prevalence of Western numbers reflects cultural contact, exposure, borrowing, and transfer through mechanisms like trade, conquest, and education. In many cases, the societies borrowing Western numbers had numbers that were similar to them; in other cases, the numbers differed, and this is one of the things that would have influenced the ease and speed with which the Western numbers were adopted. These matters would

6

Chrisomalis, 2020.

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     have been true of number systems coming into contact in ancient times as well. Western numbers differ from other cultural systems of number. As noted, they have become adopted by many contemporary societies on the planet, and they are quite old, so they have become highly elaborated, the basis for the complex mathematics that have developed in the West.7 They also tend to be what we think of when we think about what a number is. Unfortunately, we also tend to superimpose this Western idea of what a number is onto all the other numbers we encounter, regardless of whether they are Western or not, contemporary or ancient, or elaborated or not. One of the reasons for this “backward appropriation”8 – our superimposing our idea of number onto all numbers, regardless of place or time – is that we have been taught to think of number as a thing that is well defined, fixed, and timeless. This idea goes back to another of the Greek philosophers, Plato. He thought numbers were real, by which he meant abstract, immaterial, invisible, intangible, nonmental, external, and eternal entities of the same kind as those designated by words like “beauty,” “truth,” and “justice.” While no one, including Plato himself, has ever convincingly explained how we might come into contact with entities we can neither see nor touch, the idea that we somehow did has seemed to explain one of the most interesting qualities of numbers, their universality. That is, everyone has the same numbers that everyone else does, not personal or idiosyncratic systems of numbers. This is even true cross-culturally, despite the variability that is to be found there. While number is not a monolithic construct, a number is still recognizably a number, no matter how the details of its properties might differ. Numbers also work the same for everyone. If we were to add several numbers together, we would get the same results that everyone else does: 2 + 2 equals 4, assuming that everyone performs the calculation correctly. If we were to prove that an equation or 7

Gowers, 2008.

8

Rotman, 2000, p. 40.

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     mathematical statement was correct, with a mathematical proof being an argument showing that the stated assumptions of the mathematical statement logically entail its conclusion, everyone would agree that the proof was indeed evidence that the statement was true, assuming they understood it. If we were to look for prime numbers, which are numbers that can be evenly divided only by themselves and one, everyone would find the identical prime numbers. At least part of the plausibility of Platonic realism flows from this universal quality: Not only do numbers work the same way for everyone, whatever we discover about numbers and regardless of whenever or wherever we discover it, we all discover the same things and we all agree that they are the same things. Since we all discover the same things and agree that they are the same things, there is a very real sense in which numbers are “out there” somewhere, waiting for us to discover them. Another reason we superimpose our Western idea of number onto all numbers is that we can. We can because numbers are so highly similar between systems. They are so similar, we can even understand them in different notations (Roman numerals, for example) and with different bases (in the numbers used with computers, binary has a base of two, while octal and hexadecimal bases are eight and sixteen, respectively; Roman decimal numerals have a subbase of five that our Western decimal numbers lack). Numbers are so recognizable as numbers that we can even pick them out of unknown languages or scripts, like the still-untranslated writing known as Linear A used in Minos, modern Crete, about 4000 years ago,9 or the still-mostly-untranslated Proto-Elamite script used in Elam, modern Iran, about 5000 years ago.10 Numbers can also be recognizable when they are not written, which is why we understand the numerical component of the khipu, the Inka device of knotted strings. Nevertheless, identifiability as numbers depends on the degree of

9

Packard, 1974; Corazza et al., 2020.

10

Englund, 1998a, 2004.

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     numerical elaboration. Ungrouped parallel linear marks, for example, are commonly found on prehistoric artifacts, but whether they meant numbers or something else is debatable. Compare this accessibility with that of language, where a different language can be impenetrable, even when we know which one it is and even if it is written in the same alphabet we use. For example, that previous sentence, in Google Translate’s best attempt at Greek, is “Synkrínete aftín tin prosvasimótita me aftí tis glóssas, ópou mia diaforetikí glóssa boreí na eínai adiapérasti, akómi kai ótan xéroume poia eínai kai akóma ki an eínai gramméni sto ídio alfávito pou chrisimopoioúme.” While at least some combinations of letters are recognizable as meaningful and at least some words might be pronounceable because of an existing familiarity with the way the Latin alphabet works, the words themselves would not be intelligible without a knowledge of the Greek language. By comparison, it is easy to understand that the Roman numeral XVII means 17, and this understanding occurs regardless of whether we also know that the word seventeen in Latin is septemdecim. This ability to understand different number systems is asymmetric. That is, when we look at other cultural systems of number, it is from the perspective of knowing and thinking in numbers that are highly elaborated, which means that they have acquired a lot of properties over their lifespan – discreteness, ordering, relatedness, manipulability, productive grouping, and conciseness. This elaboration enables us to recognize numbers in other cultural systems, regardless of whether they have the same properties or fewer. But the converse is unlikely to be true. For example, some indigenous number systems in South America are relatively unelaborated: The numbers might count no higher than two or three, and they might not be discrete, ordered, related, manipulable, productively grouped, or concise. This means that these numbers cannot provide a similar basis for recognizing the properties of other number systems. We will look at these matters in greater detail in later chapters.

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    

     Over thousands of years, Western numbers have become quite different from their counterparts in other cultural systems. For example,  Numbers in the Western tradition are infinite, while numbers in many cultural systems are finite: They count to a certain point and then stop. Example: The Desana of the Amazonian Upper Rio Negro region count only as high as twenty.11  Western numbers are entities defined by their relations, and they are not as meaningful in isolation as they are in relation to each other, just like the relations between notes are what make music music and the relations between sounds are what make speech speech.12 In comparison, numbers in other cultural systems are equivalences or collections with fewer relations between them. Example: The Abipónes, a people who once inhabited the lowland Gran Chaco region in Argentina, expressed four as “the toes of an emu,”13 an equivalence used to exemplify collections with the property of having four members. This number followed three in counting but would not necessarily have been understood as one-more than three, two-more than two, or three-more than one.  Western numbers are discrete, while numbers in other systems can be approximate. Example: The Mundurukú of Amazonian Brazil count one, two, about three, and about four.14  Western numbers can be used to count anything, while in other cultural systems, numbers might not be used to count animate beings like people, herd animals, and deities. Different types of objects might also be counted with different numbers. Example: The Nuer of Africa know their herd animals as individuals, and so they do not count them.15 The Polynesian people of Tonga count 100 sugarcane as au, 100 coconuts as fua, 100 pieces of yam as fuhi, and 100 fish as fulu.16  Western numbers count one thing each, while in other number systems, a single number might count a pair of objects together, so that counting to ten enumerates twenty objects. Example: Tongans count many objects one by one, but they count sugarcane, coconuts, pieces of yam, and fish in pairs.17

11 14 16

12 13 Miller, 1999; Silva, 2012. Plato, 1892. Dobrizhoffer, 1822. 15 Pica & Lecomte, 2008; Rooryck et al., 2017. Evans-Pritchard, 1940. 17 Bender & Beller, 2007. Bender & Beller, 2007.

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      Western numbers include zero, a concept that many cultural systems do not include. Zero emerged relatively late in the Western cultural tradition; about 4000 years ago, it began as a blank space used to align columns of numbers to maintain their place value.18 Example: The Romans lacked a concept of zero, a characteristic for which their numerals are famous. One of the most interesting aspects of this “lack” is the idea that the number system was somehow impaired by it. On the contrary, Roman numerals were perfectly fitted to the abacus and counting boards in use at the time.19  Western numbers are grouped by tens, and such productive grouping is known as a number system’s base. Decimal organization is very common among the world’s many number systems, though number systems can also be grouped by twenty (vigesimal), five (quinary), twelve (duodecimal), four (quaternary), six (senary), and eight (octal).20 While these are not all the known bases, all of them appear to be based on the human hand. Example: The number system of the Yuki of California was organized by eights, which is understood as the effect of counting the spaces between the fingers, rather than the digits themselves, and using both hands.21  Western numbers do not have a subbase, though many number systems do. Example: Roman numerals had a base of ten (X [10], C [100], M [1000]) and a subbase of five (V [5], L [50], D [500]), making them a quinary-decimal system. The numbers of Sumer, an ancient civilization in Mesopotamia, had a base of ten and a subbase of six, giving them productive cycles of sixty, a sexagesimal system.22  Western numbers are added to produce the next higher number. Not all counting sequences add to produce the next number. Example: Some subtract: In Latin, the language of the Roman empire, nineteen (undeviginti) is one from twenty. Some overcount: In Ainu, the language of an East Asian group indigenous to Japan, twenty-six is four from ten with twenty.23 And some anticipate: In Kakoli, a language of Papua New Guinea, eighteen is two [in the next group of four above sixteen] toward twenty.24  Western numbers can be represented and/or manipulated with a variety of material forms, including the fingers, tallies, abacus beads, and written notations. While finger-counting appears to be a universal behavior – most

18 20 22 24

19 Rotman, 1987; Kaplan, 2000. Pullan, 1968; Schlimm & Neth, 2008. 21 Comrie, 2011, 2013. Dixon & Kroeber, 1907. 23 Thureau-Dangin, 1939; Lewy, 1949; Powell, 1972. Menninger, 1992. Bowers & Lepi, 1975.

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    

. . The mental number line is a construct that envisions numbers as falling along a linear continuum. Opinions are divided on whether the mental number line represents a cultural invention or an innate biological disposition. Image by the author.

societies count on their fingers – not everyone uses the same devices. Today, Western numbers are commonly represented with Hindu–Arabic notations, but this was not always the case, as we will see, and we still use all the other forms just mentioned (like tally marks: 卌) and more. Example: The Oksapmin of Papua New Guinea count to 27 using their body as a tally; traditionally, the Oksapmin did not use an abacus or notations, though Western education and currency have introduced notations and decimal organization in the last decades.25  Western numbers are often envisioned as being arranged on a linear continuum, something also known as the mental number line (MNL; see Fig. 1.3). It is an open question as to whether an MNL is innate or learned, and whether it is characteristic of all cultural systems of number. Example: Some investigations have found no evidence of the MNL in humans,26 while other scholars have found evidence of the MNL in other species.27

It is worth noting that language does not readily distinguish between any of these numbers, regardless of the properties they have. So for example, a word is translated as three regardless of whether it concerns the fuzzy Mundurukú about three; the discrete and ordinally sequenced Oksapmin three; the Polynesian three that is related

25

Saxe, 2012.

26

Núñez, 2011; Pitt et al., 2021.

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27

Rugani et al., 2015.

     to other numbers by twos, fours, and tens; or the infinitely related, notationally mediated Western three; and all the other variants described here and more. This lack of descriptiveness on the part of language has the effect of flattening the cross-cultural variability, reinforcing the impression that numbers are more similar than they actually are.

 -  Historically, the cross-cultural variability in number systems has been difficult to explain. This is because the brain has been considered to be the only place where conceptualization occurs, the braincentered or neurocentric model of numerical cognition. When the brain is considered to perform all the work in conceiving numbers, cross-cultural variability of structure and organization are taken to indicate the range of things that the brain can potentially do. Nevertheless, why the brain does things differently in some cases but not in others has been difficult to explain in the neurocentric model, particularly when some societies have many highly elaborated numbers and others very few. For reasons that are similarly unclear in the neurocentric model, at some point, the brain decides to externalize its internal mental concepts onto external material forms like tallies and notations, with these devices acting as passive recipients of that mental content. What is the alternative to the neurocentric view? The nonneurocentric model promoted in this book explains cross-cultural variation as simply the consequence of using different material forms to represent and manipulate numbers. The material forms used for these purposes are then considered to precede and inform the resultant numerical concepts and to act as an integral component of numerical thinking. In this model, the brain has less to do; rather than being responsible for all conceptualizing, its role becomes largely one of recognizing relations and patterns in the material forms used for representing and manipulating numbers. The brain remains a critical component of the cognitive system for numbers, and it is still very

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     important in an evolutionary sense because leveraging material forms for cognitive purposes to the degree that humans do is unique among animal species.28 The non-neurocentric model starts with the visual experience of quantity and symbolic notations, recognizing both as involving material forms that are engaged manuovisually – that is, by means of the hands and eyes. The model then seeks to connect the dots between these two forms. Bridging the gulf between perceptual experience and symbolic notations are devices like fingers and tallies that are also engaged manuovisually to represent and manipulate number.29 Rather than being the passive recipients of mental content, external representations have a constitutive role. Their material substance can be altered in ways that bring forth new meaning.30 Crosscultural variability in number systems – including the difference between highly elaborated and very few numbers – then becomes a relatively straightforward matter of whether material devices are used in counting, which ones are used, and how they are used.

      Numbers also change over time within any particular cultural tradition. For now, we will stick with the one we know best, the Western tradition. We need not go back as far as their Mesopotamian roots to see that they have changed a lot over time. In fact, we will look at four changes that have occurred within just the last thousand years, selected from among many changes because they are relatively easy to understand and were likely to have made a difference to the average person using numbers:  Zero became a number. About 4000 years ago, Babylonian mathematicians inserted blank spaces to align the values of columnar numbers. In India 2500 years later, these spaces became a metasign that meant the absence of any number (“no number goes here!”), and over the last 1000 years in the 28 29

Overmann & Wynn, 2019a, 2019b; Overmann, 2021f; Wynn et al., 2021. 30 Overmann, 2018a. Malafouris, 2010a.

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     West, this metasign acquired a new meaning as a sign for a number with a specified value, one less than one; a specific place in the ordinal sequence, exactly between the positive and negative integers; and unique characteristics, like its inability to divide any other number.31  One became a number. The ancient Greeks certainly did not consider one to be a number. Instead, one represented the unity, which was not just the source of all numbers: It was the source of existence itself. As for numbers, they began with plurality, which effectively started with two. Be that as it may, the mathematician Nicomachus apparently doubted whether two was a number: Just like one was the unity, two represented the dyad, another metaphysical notion.32 As recently as 1728 CE – only 300 years ago! – the encyclopedist Ephraim Chambers would observe that the status of one as a number was still a matter of debate. It did not help matters that any number multiplied by one yielded the very same number; this unique property likely reinforced the impression that one somehow differed from all the other numbers.33

’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[34] is very angry with the Maintainers of this 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. (Chambers, 1728, p. 323)

 Hindu–Arabic notations (0 through 9) replaced the Roman ones (I, II, III. . .). This transition was neither easy nor quick. Merchants and bankers were initially suspicious of the new notations, particularly of zero, since it seemed to make falsifying values far too easy: 10 could become 100 by simply adding another of the dodgy signs.35 In comparison, Roman 31 33 34

35

32 Rotman, 1987; Kaplan, 2000. Nicomachus, 1926; Evans, 1977. Nicomachus, 1926. Stevinus refers to the mathematician Simon Stevin, who helped influence the reconceptualization of one as a number in the sixteenth century CE. Ifrah, 1985; Rotman, 1987; Kaplan, 2000.

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     numerals were harder to falsify, as it would be impossible to turn X [10] into C [100] by adding a zero.  Calculating with algorithms involving learned relations, mental judgments, and handwritten notations supplanted the mechanical exchange of values on abaci and counting boards. This transition too was neither easy nor quick. The debate over whether it was better to calculate by means of knowledge-based algorithms, rather than by moving beads on an abacus or the counters known as jettons on a counting board, took centuries to resolve. The transition also involved contests like the one shown in Fig. 1.4, something that has persisted to recent decades as contests of speed and accuracy between the abacus and electronic calculators or computers.36

If we look further back in time, say, another thousand years or so, Greek philosophical ideas about numbers and other matters were even more influential. The idea of zero was inhibited by metaphysical ideas about being (existence, which was good) and non-being (nonexistence, which was a horrifying possibility).37 “Irrational” numbers like π and √2, which we know today as fractions that neither terminate nor repeat, so greatly challenged conceptions of what numbers were and how they were supposed to behave that the mathematician Hippasus, possibly their discoverer, is said to have drowned. Reports of the incident differ greatly regarding what really happened. Some accounts have Hippasus punished by the gods for impious behavior; others say his fellow mathematicians did him in. Some say he perished because he divulged the existence of irrational numbers, others because he told the secret of how they might be calculated to men who were not initiates of the philosophers’ guild. One account has Hippasus throwing himself into the sea, driven incurably mad by the irrational nature of the numbers in question. Considering that numbers were foundational to Greek concepts of existence, the idea that numerical irrationality could be so thoroughly confounding at least has the virtue of consistency, if a trifle over-exacting of enthusiastic devotion.

36

Pullan, 1968; Stone, 1972; Evans, 1977; Reynolds, 1993.

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37

Rotman, 1987.

    

. . Woodcut from Margarita Philosophica [Pearl of Wisdom], originally published in 1503 as one of the first printed encyclopedias of general knowledge. To the left, the Roman mathematician Boethius calculates with algorithms and notations, while to the right, the Greek philosopher Pythagoras uses a counting board. Arithmetic personified as a lady looks on, turned to the left apparently to favor the algorithmic approach. Image in the public domain.

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    

    ,   ,     ? Despite all this variability between cultural traditions and within any particular tradition over time, numbers are astonishingly similar. For one thing, they all have the same cardinality. For another, they all have the same ordinal sequencing and ordering by increasing magnitude. Thirdly, each next number increases over the previous number by one, though the way in which each next number is derived can be quite variable, as shown earlier with examples from Latin, Ainu, and Kakoli. And all numbers demonstrate the same patterns: For example, the prime numbers are the prime numbers for everyone. Another of those patterns is an anatomically derived base number. The ten in decimal comes from the fingers of both hands; the twenty in vigesimal comes from all the fingers and toes; and the five in quinary comes from the fingers of one hand.38 A base of four (quaternary) might indicate counting the spaces between the fingers, rather than the digits themselves,39 or considering the fingers separately (counted) from the thumb (not counted). A base of eight (octal) might simply double the method of counting by fours, much as decimal doubles quinary. A base of six (senary) might emerge from including the thumb joint along with the fingers, with twelve (duodecimal) emerging from its doubling or from using the three segments on each of the four fingers while omitting the thumb. A base of fourteen uses the three segments per finger and includes the thumb with its two segments. As for sexagesimal or the base sixty number system of Mesopotamia, no one really knows the reason for it. It had a base of ten and a subbase of six. While ten most likely was related to the fingers,40 the reason for six as the next higher base has simply been lost to the passage of time. Why consider all this variability between cultural traditions and within any particular cultural tradition over time? While we are

38

Epps, 2006.

39

Dixon & Kroeber, 1907.

40

Overmann, 2019b.

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     interested in what changes, we are even more interested in why and how things change, as these can illuminate aspects of the processes whereby numbers emerge and become elaborated. And in the process of learning about these things, we will get an overview of how numbers work for human societies in all sorts of places and at a lot of different times: contemporary, historical, and prehistoric.

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

Converging Perspectives on Numbers

Numbers can be studied from a lot of different perspectives. This is both part of their fascination and a unique interdisciplinary challenge, since few of those perspectives incorporate insights from the other fields that also study numbers. This leaves the endeavor a bit fractured and disconnected, bringing to mind the old Buddhist parable about the blind men and the elephant: Everyone has a bit of solid evidence, and no one quite has the big picture. Assembling that picture requires sorting through the data generated

by

psychologists,

neuroscientists,

paleoneurologists,

zoologists, linguists, ethnographers, archaeologists, philosophers, mathematicians, and historians, and then seeing how everything might fit together within a holistic explanatory framework. Cumulatively, these data provide a lot of pieces of a large and complex unknown. Not all of these pieces might fit because not all of them may be equally pertinent to the question of numerical origins. For example, a significant portion of the psychological literature focuses on how children typically acquire numbers as they mature, including characteristics of numerical language like grammatical number, the way we mark singularity and plurality in nouns, that have been shown to aid the developmental acquisition of number concepts. This assumes both a society in which numbers are already available to be learned and a language with the specific helpful feature. Be that as it may, not all societies have numbers, and not all languages have the helpful feature, and some societies whose language lacks the helpful feature are nonetheless highly numerate and easily acquire numerical concepts without it. These matters suggest that the developmental data, and for that matter, the 

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     linguistic data on helpful grammatical features, are less immediate to the question of how numbers originate, compared to data on how we perceive quantity. Constructing a holistic explanation of how things work from the data that seem useful is a challenge. The data are generally disconnected in space and time, and they span a multitude of disciplines with diverse methods and foundational assumptions. To bridge these gaps, we depend on inference: using the data we have to explain what we cannot directly observe. To be successful, the data must be pertinent to the phenomenon being explained.1 A convincing inference will also be supported by multiple lines of evidence. An example is finger-counting, a behavior found in emerging number systems, highly elaborated number systems, and number systems whose elaboration is intermediate between the two. The fact that finger-counting is found in emerging systems implies that it would have been used in the emerging stage of elaborated systems as well, though this cannot be observed because of the passage of time. The idea that finger-counting was likely present in earlier stages of elaborated systems can be corroborated by certain features of language and structure: for example, terms that mean both “finger” and “digit,” and using five or ten as a productive base.2 Finally, the observation that most societies count on their fingers can be explained through neurological interconnections and interactions between the parts of the human brain that perceive quantity and “know” the fingers.3 Inferential arguments are common in evolutionary studies, especially those that investigate our cognitive origins. They will not satisfy the historian, who wants all the bits and pieces drawn from documents. Nor will they satisfy the psychologist or cognitive scientist, who prefer what can be tested in the lab. Since these techniques

1 3

2 Botha, 2006, 2016. Epps et al., 2012. Roux et al., 2003; Penner-Wilger et al., 2007; Grabner et al., 2009, 2013; Reeve & Humberstone, 2011.

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     are not possible with ancient populations, establishing that they had, for example, the same sense of number that we do is instead inferred from the facts that the numerical features of their languages are patterned in the same way that ours are, and ours are patterned by numerosity.4 In whatever way we put the big picture together, we will be left with some really big questions to answer: What are numbers? How and why do they work the way they do? When did they begin? And where do they come from? We will look at these questions by starting with classical, historical, and recent ideas about how numbers originate, including realism, intuitionism, and nativist, linguistic, embodied, and extended models of numeracy.

       No one really knows how old numbers are, or why they might have started. The numbers in the Western tradition – the numbers that most readers will likely know and use – are at least 6000 years old, and possibly as old as 30,000 or even 100,000 years or more. However old numbers might be, older and younger numbers do not have the same properties, even when they are historically related, and this poses the even more challenging questions of how and why numbers began and why they differ. Our ideas about what numbers are and where they come from go back to the ancient Greeks, particularly Plato. The Greeks are the first mathematicians thought to have been interested in numbers for the sake of what they were, not just for what they could do.5 Plato said numbers were universals, or abstract, repeatable properties that existed as ideal forms, independently of the physical things that exemplified and instantiated them in the world or the minds that apprehended them.6 Universals were held to exist – to be real – in some way that is still rather poorly defined, despite the several

4

Overmann, 2019b.

5

Høyrup, 1990.

6

Linnebo, 2018.

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     millennia of philosophical energy that has been 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. 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 societies with different cultures, priorities and needs, and environmental contexts. Another is that implicit to those numbers are the very same relations and patterns within the relations, like the prime numbers. As the linguist Noam Chomsky once put it, this elevates numerical and mathematical properties to facts, and the facts to truth:

We might say that in the study of arithmetic there are just the arithmetical truths, there are the properties of the numbers. So the fact that the square root of two is irrational is just like the fact that the sun has helium in it. And if we take a classical view of mathematics, then the truths of mathematics are just truths on a par with the truths of physics even. They are truths that are independent of our ways of thinking. So the theory of mathematics in a sense is independent of any particular techniques that we use for carrying out arithmetical computations or making mathematical constructions. (Chomsky, 2004b, p. 43)

People not only find the same relations and patterns in a counting sequence, they also always get the same result when multiplying or dividing this by that. They will also agree, with unusual conviction, that the right answer is the right answer, the intersubjective verifiability noted by the German mathematician and logician Gottlob Frege.7 That is, the truths of numbers and mathematics are

7

Frege, 1956.

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     not personal. We share them, and we apprehend them in such a way that we agree we share them. This, too, seems to make numbers things that exist externally to ourselves.

[Do] numbers exist in the thinking subject, or in the objects which give rise to them, or in both[?] In the spatial sense they are, in any case, neither inside nor outside either the subject or any object. But, of course, they are outside the subject in the sense that they are not subjective. (Frege, 1956, p. 105)

Given these qualities, the idea that we discover numbers seems plausible, if not correct. As things that are discovered do seem to need to exist in order to be discovered, discoverability gives numbers an air of existing externally 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 do not change, but our familiarity and understanding improve over time. Some hint that we might not be entirely satisfied with the classical realist view lies in the continuing debate. Were realism a satisfactory answer to the questions of what numbers are and where they come from, one might think that 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 that they are purely mental constructs. Mathematics becomes “the study of mental constructions of a certain type.”8 Taking this view, the Dutch mathematician Luitzen Egbertus Jan Brouwer proposed that number concepts were realized by “the consciousness of self in time,”9 the view of numerical origins known as intuitionism. Brouwer “roughly sketched” his view of

8

Chomsky, 2004b, p. 43.

9

Brouwer, 1981, p. 90.

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     how we intuit numbers through a somewhat mystical “inner experience”:

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. (Brouwer, 1981, p. 90)

In Brouwer’s view, the mind apprehends twoness (or “twoity,” as he called it), which gives rise to an idea of oneness (or unity); when added together, these in turn give rise to the notion of threeness, and all the other numbers and their mathematical manipulations unfold in the mind from there. Interestingly, the data on numbers in language show that names for one and two emerge first, before names for any other numbers, and that terms for three can and do emerge as one and two or two and one. Nevertheless, the study of cultural number systems shows that two appears to be just as likely to emerge before one as the other way around,10 and that these initial terms do not come preloaded with all the properties – ordering, manipulability, relations, conciseness – needed for mathematical manipulation. Given that he crafted an alternative, Brouwer was likely dissatisfied by the realist answer. He also wanted to establish a secure foundation for mathematical truth. Foundationalism 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 (truth-preserving), which meant mathematical

10

Closs, 1993.

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     conclusions would only be sound if they were anchored to a bedrock of truth. He chose numbers for this foundation, likely because they have been recognized as the basis of mathematics for so long that their position as such is definitional. He was also aware that previous foundationalist attempts had failed to derive all of mathematics from logic and sets, sometimes spectacularly so.11 His choice of a starting point would therefore be 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 over time. His notions that numbers originated as mental content and that this origin gave them the requisite foundational solidity came from “the old metaphysical view that individual consciousness is the one and only source of knowledge.”12 This view of mental content has a long and distinguished history. In the seventeenth century, the French philosopher René Descartes used it to establish a foundation for knowledge generally, proving that he himself existed, and so did the world and God, and what he knew of both emerged through his access to his own mental content: He perceived his thoughts, so he must exist, and so must the world and God; he could not be deceived about what he perceived because God existed and was good.13 “I think, therefore I am,” and the rest follows. Access to mental content, unfortunately, is not nearly as infallible as Descartes and Brouwer wanted to believe. Much, if not most, of our mental content is unconscious, and hence, inaccessible to introspection. The conscious part is altered when and because we introspect in a way that is imperfectly analogous to the uncertainty principle in quantum mechanics, and this alteration degrades the accuracy of what we think we find by introspecting. We may then misunderstand, misinterpret, or misreport what we think we find, making our reports of already inaccurate mental content unreliable. Reporting is also highly individual, and there are few criteria for

11 13

Tennant, 2017; Irvine & Deutsch, 2020. Descartes, 1637, 1664, 1993.

12

Ferreirós, 2008, pp. 148–149.

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     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 founding mathematical certainty and truth. Because of these problems, the label “introspectionist” is not a compliment within the field of psychology. 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 seemed to him in his. He was born and enculturated into a society whose numbers were both highly elaborated and deeply interwoven into the cultural and linguistic fabric. Beyond an impressive mathematical talent, his vocation as a 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 his time. Cross-cultural variability in number systems was known, but during Brouwer’s life and even toward the end of the twentieth century, such variability was attributed to nineteenth-century notions about different societal modes of thinking, associated today with scholars like Jean Piaget and Lucian Lévy-Bruhl.14 Concrete thinking was what so-called primitive societies did, while Brouwer’s own, presumably advanced, enlightened, rational, and scientific culture had abstract thinking. Societies with fewer numbers than Brouwer’s own were, baldly stated, considered less than human, a characterization that has continued to permeate scholarly discourse into uncomfortably recent times, if in a weakened or disguised form. For example, Chomsky has positioned the failure to generate numbers as a cognitive deficiency: 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 14

Lévy-Bruhl, 1910, 1927; Piaget, 1928, 1952.

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     capacity that has obvious selection value, should be latent and not used. That would be amazing if it were true.”15 Another linguist, James Hurford, has stated 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.”16 Interpretations of these statements range from the trite – obviously, when peoples do not have numbers, whatever causes numbers to emerge and elaborate did not do so – to the racist, positioning peoples without numbers as “biologically less than truly human.”17 Even if his view of numerical origins were mistaken, Brouwer might have been inclined to defend it. If numbers were real or constructed like he thought mathematics were, rather than introspectively intuited as he claimed, the crucial foundation of his own mathematical system would have been kicked right out from underneath him. Brouwer 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 and material devices behave in numbers. This evidence that Brouwer did not have and could not have known is the subject of the next several chapters. A broader perspective on Brouwer sees him as viewing numbers and our capacity for understanding them as reflections and products of the human mind and the universe it inhabits. This we can agree with, even as we continue to disagree with Brouwer on the details of how numbers emerge and elaborate. We will also disagree with him regarding the nature of the human mind, since Brouwer most likely, given his time and introspectionist bent, conceived the mind in neurocentric terms. And we might be left to wonder whether the material origin for numbers sketched out here explains anything about some of

15 17

16 Chomsky, 1982, pp. 18–19. Hurford, 1987, p. 73. Verran, 2000a, p. 292, critiquing Chomsky (1980, 1982).

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     the peculiar properties that numbers have as concepts, created as they are by an extended human mind whose boundaries encompass material forms, and anchored as they are by material forms whose properties become numerical properties.

      More recent models of numerical origins come from, respectively, psychology, linguistics, cognitive science, and archaeology informed by neuroscience and the philosophy of mind.18 The first model is nativism, the idea that the biological endowment for appreciating quantity – numerosity – is the basis for numbers, in humans and animals alike.19 The second, the linguistic model, sees numbers as a function or component of language, which would make numbers unique to humans as the only species with language.20 Undoubtedly, both numerosity and language contribute to numeracy, but it is not at all certain that either – or even both together – is all we need, since not all human societies develop numbers, despite the fact that all humans have the biological abilities for appreciating quantity and communicating in language. A third model, the embodied model proposed by the linguist George Lakoff and the cognitive scientist Raphael Núñez,21 adds to numerosity and language our embodied experience of objects as collections and our ability to form and express metaphorical concepts through language. While conceptualization is still envisioned to be thoroughly mental, it is nonetheless recognized as embodied and embedded; this arguably lets the mind start to “leak out” of the brain. A fourth model – the extended approach developed here within the framework of Material Engagement Theory22 and the only one of the four that is non-neurocentric23 – adds the material devices used for representing and manipulating numbers to the cognitive system for numbers.

18 20 22

19 Overmann, 2021e. Dehaene, 2011; Nieder, 2017a, 2017b. 21 Hurford, 1987, 2007; Chomsky, 2004a, 2004b. Lakoff & Núñez, 2000. 23 Malafouris, 2013. Malafouris, 2013; Overmann, 2019c.

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     Of course, readers need not adopt any particular position on the origin of numbers. Some familiarity with the issues, along with the ability to recognize the strengths and shortfalls of the respective positions, will aid in following the arguments.

A quick summary of the origin hypotheses and models 24  Realism: Numbers exist, and we discover them. Support: All societies

discover the “same” numbers. Problems: Are all numbers really the “same,” considered cross-culturally or over time? How would we discover or interact with universals, entities that are intangible and invisible?  Intuitionism: Numbers are intuitive and purely mental constructs; they exist in the mind, and we just need to look inside the mind to discover them.25 Support: We do have an innate capacity for numbers. Problems: Introspectionist. Is the innate ability to appreciate quantity the same thing as having a number concept? Does not account for why or how we use material structures for counting.  Nativist model: Numbers are biologically endowed.26 Support: The innate capacity to appreciate quantity is shared by many species and influences numeracy as it is expressed in humans. Problems: Do bees really have the same concept of zero that humans do? What do language and material culture contribute?  Linguistic model: Numbers are a subset of language.27 Support: Numbers have regularities and language has regularities. Problems: Do spoken numbers, material numbers, and language all have the same regularities? If all humans have language but not all societies have numbers, what does this imply about societies that lack numbers, or language as the source of numbers?  Embodied model: Numbers come from our experience of collections of objects, conceptual blending, and metaphoric expression in language.28 Support: Highly plausible in drawing on numerosity, language, conceptualization, and the experience of objects with quantity. Problem: Does not consider data on emerging number systems, nor the role of material objects in numerical conceptualization and thinking.

24 27

25 26 Maddy, 1990. Brouwer, 1981. Nieder, 2017a, 2017b. 28 Chomsky, 2004a, 2004b. Lakoff & Núñez, 2000.

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      Extended model: Quantity perception is an integral system comprised of brain, body, and world; the material dimension is subject to alteration in ways that bring forth meaning, ultimately yielding concepts of number.29 Support: Holistic view of the interdisciplinary data. Problem: Takes the radical view that cognition and mind are not just the brain but include material objects and the body. Takes an unconventional view of what constitutes a material device for numbers by including things like fingers, sorting behaviors and ephemeral structures, and symbols.

        If numbers are real but immaterial, as realism claims, or concepts that emerge from the brain, as intuitionism and nativism hold, or words and ideas generated by our cognitive capacity for language, as linguists and embodied modelers believe, how might we investigate these things? One way is by looking at the numbers of different societies and cultures. These exhibit an astonishing variety. Some systems, like the Western tradition, are very old and highly elaborated; other systems, like those of indigenous peoples in Australia and South America, are currently emerging and relatively unelaborated. One problem in this research is that contact between Westerners and indigenous peoples spreads concepts and language, particularly those of numbers, which transfer with unusual speed. The result is that Western numbers quickly displace indigenous numerical concepts, behaviors, and language. This means that the best insight into what precontact numbers were like often comes from the historical ethnographic literature. When the historical ethnographic literature is consulted to see how different people counted, two things are immediately apparent. First, the descriptions are remarkably consistent, no matter how widely separated they are in place and time. The Norwegian mathematician Øystein Ore, for example, began his book on number theory by 29

Malafouris, 2013; Overmann, 2019b.

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     observing that most, if not all, societies are known to use their fingers and other material devices for counting:

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. (Ore, 1948, p. 1)

The second thing that is immediately apparent is that the historical ethnographic literature is rife with biases and misunderstandings, and it is sloppy – to say the least – in how it identifies societies and cultural traditions. Many of the names used, for example, cannot easily be traced to specific contemporary societies or cultural groups, and for those that can, the descriptions are not necessarily faithful to the ways in which a particular society or group might understand its own cultural and historical traditions. As noted, however, these observations are often the only insight we have into what precontact numbers were like. Further, their consistency across and despite widely divergent circumstances – times, places, observers, training, societies, cultures, contexts, purposes, agendas – gives these data a certain validity and reliability. Simply, while individual observations are potentially unreliable, the fact that they report the same thing, time and time again under vastly different circumstances, means that cumulatively, they plausibly represent what is actually the case. This is why the historical ethnographic literature continues to be a valuable resource for insight into pre- and early-contact number systems, despite the various problems associated with it.

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     More recent perspectives on numbers are also problematic. An example is the work of Jean Piaget, the Swiss clinical psychologist best known for his work on cognitive development in children. Noting differences between childish thinking and adult thought, Piaget envisioned a four-stage progression from one to the other.30 Children from birth to the age of two were in the sensorimotor stage, experiencing the world through sensation and movement. Years two to seven 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 seven to twelve 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 twelve 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 conceptual acquisition as a process of developing biologically predetermined cognitive structures through the experience of objects. 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,31 similar to the way that language proficiency is developed through exposure to radically different sets of utterances. While further elaboration of Piagetian developmental theory is tangential to the present argument, it is important to mention two interrelated matters. The first concerns the implications of applying developmental theories like Piaget’s to entire societies; the second is how the ontogenetic aspects of conceptual acquisition might overlap with the historical development of concepts. To extend his ideas about cognitive ontogenesis in children to entire societies, Piaget drew upon earlier work by sociologist 30

Piaget, 1952; Inhelder & Piaget, 1958.

31

Nicolopoulou, 1997, p. 207.

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     Lucien Lévy-Bruhl.32 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, which consisted 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 nineteenth century; they continued well into the twentieth century and, as was earlier shown in discussing the idea that numbers originate in language, to uncomfortably recent times as well, albeit in weaker form. Another example is the construal of an “abstract–concrete distinction” in Mesopotamian and other number systems,33 an idea based on the work of Piaget and LévyBruhl.34 Piaget construed many parallels between the so-called primitive mentality of the “civilisations inférieures” and the immature “mentalité enfantine”:35 Examples include the tendency to assert without proof, the affective character of thought, its global, nonanalytical 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. (Piaget, 1928, p. 194, as translated)

32 33 34

Lévy-Bruhl, 1910, 1927. E.g., as found in the work of Damerow (1996b) and Schmandt-Besserat (1992a). 35 As noted in Overmann, 2021b. Piaget, 1928, pp. 191–201.

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     Though using the unfortunately biased language of his time, Piaget was attempting to describe real phenomena. There are in fact meaningful conceptual differences between societies, as for example, across different systems of religion and science. Obviously, none warrants the labels applied in nineteenth-century discourse, since societies do not develop into adults like children do. Numbers too differ between cultural systems in terms of their properties and how they are used. And contemporary psychology is only now beginning to recognize and appreciate cognitive differences between individuals from societies that are Western, educated, industrialized, rich, and democratic – or WEIRD – and societies that are not WEIRD.36 This distinction is an important one because WEIRD (or Western) people are only a fraction of the world’s population, yet they skew our understanding of what human cognition is like because Western scientists perform most of the experiments published in the psychological literature, and most of these experiments involve Western participants. Some of the differences between WEIRD and non-WEIRD people are surprising. One is demonstrated with the Müller-Lyer illusion (Fig. 2.1), where lines of identical length are perceived as longer or shorter depending on whether their end arrows point out or in. Western people are reportedly more prone to seeing lines with unequal length, an effect on visual perception thought related to a habituation to urban linearity.37 That is, cognitive differences reflect enculturation into material differences, making the human brain “as much a cultural artifact as a biological entity.”38 For numbers, brains enculturated into different cultural systems appear to process identical symbolic notations differently, a phenomenon language cannot fully explain.39 Significantly, the ability to perceive nonsymbolic quantity appears impervious to the WEIRD/non-WEIRD divide, giving all cultures the identical starting point for numbers.40 36 38 40

37 Henrich et al., 2010. Henrich et al., 2010, pp. 64–65. 39 Malafouris, 2013, p. 45. Tang et al., 2006. Gordon, 2004; Butterworth et al., 2008; Henrich et al., 2010, 2011; Rooryck, et al., 2017.

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     . . The Müller-Lyer illusion. Western people tend to see the lines as unequal, where in fact they have the same length. Image in the public domain.

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 indeed plausible similarities between this process and ontogenetic development. For example, both involve interacting with social others and the physical world. Both are progressive in the sense that later constructions or stages 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 as the result 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 just 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 and availability of concepts, terminology, and knowledgeable others, and its social invention, which presumes that none of these things are available. Arguably, 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 – since

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     they do not yet have the cognitive wherewithal to understand the concepts, it seems unlikely they would be able to invent new ones. Nevertheless, 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. That is, putting a concept into terms a child can understand helps to clarify it in the mind of the adult who explains it, and this clarity becomes part of the conceptual knowledge of both. 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 in developmental psychology 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. We will not spend any more time on the developmental aspects of numbers, since they are not reasonably significant to numerical origins, 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. 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.41 In any case, it is tangential, if not orthogonal, to the present discussion of how numbers are realized and elaborated, for the reasons mentioned. By the 1980s, cultural studies were grappling with the question of how to characterize the differences between non-Western and Western numbers. Non-Western numbers, which had been deemed “non-scientific” by authors like Piaget and Lévy-Bruhl, were 41

Oesterdiekhoff, 2016.

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     redesignated as “scientific” – albeit qualitatively different from “science” in the Western sense42 – implying a continuing, though weaker, form of Piaget’s distinction. How best to characterize crosscultural differences in numbers and numerical thinking remains an open issue today, one that can be at least partly resolved by recognizing material forms and associated properties as their basis. Some of these issues will surface again when we look at the contemporary literature on number systems. We will not solve these issues here, nor is solving them the goal of this book. An awareness of the different perspectives and biases, especially with regard to the historical literature on cultural number systems, is critical.

      When the material component of numbers is examined to understand how numbers emerge and elaborate, archaeology becomes pertinent to the inquiry as the science that studies material objects. Archaeological evidence of devices used for counting, past and present, can provide insight into numerical prehistory at a depth of time that linguistic techniques generally cannot touch. While the lexical and syntactic comparisons involved in reconstructing protolanguages, including some words for numbers, currently reach back 10,000 or 15,000 years, some artifacts possibly used for counting are tens of thousands of years older, dated to the European Palaeolithic and African Middle Stone Age. A more recent linguistic technique assesses the rate of change in small number-words (up to five) as unusually slow, admitting the possibility that in some language families, number-words may be as old as 100,000 years or even older.43 This technique, unfortunately, does not establish an age for numbers, nor does it pinpoint when numbers emerged. Neither does it acknowledge the likelihood that short number-words like five would have been preceded by lengthy phrases – like all the fingers on one hand, 42

E.g., Verran, 2001.

43

Pagel et al., 2013; Pagel & Meade, 2017; Calude, 2021.

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     found in emerging systems today – that imply the potential for numerical emergence to have occurred at an even greater depth of time. Admittedly,

archaeology

cannot

pinpoint

exactly

when

numbers first emerged either. A variety of material devices that look like the sort of things that could possibly have been used for numbers have been dated to the past 100,000 years or so. Discerning whether notched bones or strung beads were actually used to count, rather than meaning merely that someone once whittled or wore a necklace, is a continuing challenge. This has motivated archaeologists like Francesco d’Errico and Brian Hayden to offer methods and criteria intended to remove at least some of the ambiguity from determining purpose and use in artifactual forms that look like tallies.44 Chapter 11 will examine how successful these techniques might actually be, as well as what archaeology is likely to discover – or miss – in the emergence and elaboration of numbers as discerned from the prehistoric record. 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? Historically, archaeologists – along with pretty much everyone else in all the other disciplines – have answered the first question by positioning physical devices as external memory storage, something that aids our cognition by offloading our mental content. In this neurocentric framework, memory being stored externally means that the brain has an idea that is then marked on a physical form in a way that represents and preserves the idea materially. This means that the brain does all the work, and its job is made easier when the device assists what the brain does – once the brain has formed the concept and thought of how to make the device that holds it. The device itself is passive. However, this is far from the only way to think about what material devices do in numbers, a matter that we will explore in depth in later chapters. 44

D’Errico, 1991; d’Errico & Cacho, 1994; Hayden, 2021.

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

The Brain in Numbers

Numbers involve various functions, capacities, regions, and connections of the brain. Here we will focus on those important to understanding how numbers emerge and become elaborated, particularly through the use of material forms but also in regard to spoken forms of numbers:  Numerosity is the innate sense of quantity that humans share with many other species. In humans and nonhuman primates, numerosity is a function of the intraparietal sulcus, a region of the parietal lobe. Numerosity governs what we can and cannot see, quantity-wise, and this influences both our need to use material forms and how we use them.  Categorizing is the ability that groups or differentiates objects according to the similarities or dissimilarities of their properties, relations, or functions, while abstraction is the process of deriving general concepts and rules from specific properties, relations, or functions. Small sets of objects – singles and pairs – have quantities that are perceptible, and the similarities and dissimilarities of these properties as shared between sets are the plausible basis for concepts of one and two. These concepts are then expressed materially through the fingers, or verbally by describing or naming an object that exemplifies the quantity.  The mental number line (MNL) is the ability to conceptualize numbers as arranged along a linear continuum. The MNL might be an innate tendency for representational structure that influences numerical conceptualization and expression, or it might be an effect of interacting with material forms like writing, a debate that is currently unsettled in the literature.

Language, another human capacity involving the brain, is mentioned here as it relates to the brain and numerosity and discussed further in Chapter 5 in terms of vocabulary and structure. While language has been proposed as the reason we have numbers, it 

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     involves neuroanatomy and neural activation patterns distinct from those observed in numbers. Language preferentially involves the frontal and temporal lobes and is primarily an auditory phenomenon,1 while numeracy, the ability to reason with numbers, and numerosity, the perceptual system for quantity, are associated with activity in the parietal lobes and are primarily visual phenomena.2 These differences pose the question of just how closely related language and numeracy are from the standpoint of the brain. They are, in fact, doubly dissociable, the gold standard in neuropsychology for establishing independence of brain form and function. When strokes or injuries damage particular regions of the human brain, the resultant impairments can be used to associate brain structures with brain functions. Doubly dissociable means that damage to regions associated with language impairs the abilities to produce and comprehend speech without producing concomitant impairments in numeracy, and conversely, that damage to regions associated with numeracy impairs the abilities to understand and use numbers without producing concomitant impairments in language.3 However distinct and independent they are in terms of brain form and function, language and numeracy are also mutually interacting and coinfluential functions, as we will see in examining the effects of numerosity on numerical language. Three others psychological capacities are worth briefly mentioning. Working memory is the ability to hold and manipulate information mentally, in conscious awareness.4 Attention is the ability to direct cognitive resources in a flexible manner with components of “awareness, vigilance, saliency, executive control, and learning.”5 Object tracking is a cognitive resource that enables us to attend to distinct objects and stimuli in the environment.6 All of these are finite

1 2 3 4 5

Tremblay & Dick, 2016. Orban et al., 2006; Fias et al., 2007; Amalric & Dehaene, 2016, 2018. Brannon, 2005; Varley et al., 2005; Monti et al., 2012; Amalric & Dehaene, 2016. Baddeley & Hitch, 1974; Baddeley & Logie, 1999; Baddeley, 2007. 6 Lindsay, 2020, p. 1. Kahneman et al., 1992; Carey, 2009.

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     in their resources, which are limited to about three or four items.7 These resources are thought to underlie and limit numerosity.

,      Numerosity is the perceptual experience of quantity. In humans, numerosity is functionally divided into subitization, the ability to rapidly and unambiguously recognize quantities up to about three or four, and magnitude appreciation, the ability to appreciate bigger and smaller in quantities above about three or four (Fig. 3.1). These functions underlie how numbers are realized, expressed, and elaborated, particularly in material forms because of the use of vision in engaging them, but also in language. Subitization is constrained by memory, attention, and/or object tracking,8 resources that limit the number of objects or items we can simultaneously attend or keep track of. In turn, subitization patterns how numbers first emerge, a function of our ability to see and

. . 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 discern bigger from smaller in the quantity of groups, as long as the difference is above a threshold of noticeability. Without counting, the quantity on the right, which exceeds the subitizing limit, is just many. Image by the author. Previously published in Overmann (2019b, Fig. 4.1, p. 44).

7

8

Bravo & Nakayama, 1992; Wolfe & Horowitz, 2004; Egeth et al., 2008; Olivers & Watson, 2008; Railo et al., 2008; Burr et al., 2010; Cowan, 2010; Charlton & Starkey, 2011. Olivers & Watson, 2008; Railo et al., 2008; Carey, 2009; Burr et al., 2010; Ester et al., 2012; Rooryck et al., 2017.

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     appreciate small quantities but not those exceeding the subitizing range. Subitization also informs why and how we use material devices, initially to express perceptible quantities and then to access and organize quantities beyond the subitizing range. Subitization also influences the expression of quantity in language, as well as how often number-words are used in language, matters important to understanding how numbers emerge and why we can be sure ancient people had the same ability to perceive quantity that we do. Consistent with subitization, the first numbers to emerge are one and two, sometimes three, and occasionally four; quantities above the subitizing range are called many. Consistent with magnitude appreciation, many is often subdivided as big many and small many. These correspondences between the way that numerosity functions and the way that numbers emerge appear consistently and reliably across languages, cultures, times, and places, and this demonstrates that the perceptual experience of quantity informs the initial expression of numbers.9 In essence, numbers begin as concepts of the quantities we can see. Given how the perceptual system for quantity functions, discrete quantities above the subitizing range necessarily emerge from the undifferentiated many through interactions with material forms – often, initially, the hand for the quantities five and ten. Material forms act to scaffold quantities beyond the subitizing range, enabling them to crystalize as distinct manuovisual entities. Numerosity continues to influence the material forms used as numbers become elaborated. For example, nonsubitizable amounts of notches on a tally or elements in a notation are visually indistinguishable as discrete quantities, and this motivates the use of grouping, typically in subitizable amounts (particularly threes) and/or in bundles based on the hands (e.g., fives, tens, or twenties). These matters will be discussed in detail in later chapters.

9

Overmann, 2015; Rinaldi & Marelli, 2020.

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     Numerosity also structures how numbers are expressed and used in language. Not only do the initial lexical or verbal forms conform to the perceptual experience of quantity – one, two, three, many – so too does grammatical number, a feature found in some languages that distinguishes singulars, sometimes duals, and perhaps, occasionally trials.10 In languages with well-established number systems, the most frequently used numbers in speech are the subitizable ones, with numbers higher than four showing a rapid drop-off in their frequency of use (Fig. 3.2). Across languages, this is true regardless of whether the number-terms in question are lexical, ordinal (forms of number that order: first, second, third. . .), or grammatical. In comparison, color terms, another vocabulary associated with a perceptual system that exhibits cross-cultural tendencies in the order in which terms emerge in language,11 do not drop off in their frequency of use above the subitizing range. For its part, magnitude appreciation discerns differences in quantity large enough to be perceived, a threshold of noticeability governed by Weber’s Law (or the Weber–Fechner constant).12 That is, to be perceptible, the difference must be large enough for us to discern it. This is true not just for quantity, but also for perceptual modalities like temperature, weight, size, length, angle, loudness, pitch, brightness, and color. For example, if we were to hold two objects, one in each hand, their temperatures or weights would need to differ sufficiently for us to detect a difference; when they do not differ sufficiently, we cannot tell that one is cooler or lighter, the other warmer or heavier. Similarly, when we look at two groups of objects, the difference in their quantity must fall above the threshold of noticeability for us to be able to tell them apart as bigger and smaller. For quantity, noticeability is also influenced by object characteristics like size and proximity; in experiments, researchers control

10 11 12

Overmann, 2015; also see Franzon et al., 2019. Berlin & Kay, 1969; Kay et al., 1991. Weber, 1834; Fechner, 1860; also see Masin, 2009.

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    

. . Frequency of use for the lexical numbers one through twenty in English (top left), Chinese (top right), and Arabic (bottom left), selected because they represent different language families and are also associated with electronic corpora suitable for analysis. Ordinal numbers exhibit similar frequencies (Fig. 5.1). The subitizable numbers are the most frequently used; above this range, usage frequency drops off in a way that evokes the Weber–Fechner constant (bottom right). Small increases at five and ten reflect usage consistent with decimal organization. The data were drawn from the British National Corpus (https://corpus.byu.edu/bnc) and University of Leipzig Corpora Collection (https://corpora.uni-leipzig.de). Image by the author.

these characteristics in order to maintain a focus on quantity discrimination. Subitization and magnitude appreciation can be tested with nonsymbolic quantity, the kind of quantity shown in Fig. 3.1 that both humans and animals can appreciate. Humans can also appreciate and be tested with numbers in symbolic forms, assuming that they

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     live in cultures where such things exist and have had the opportunity to learn them. In the symbolic form of numerals, the quantities shown in Fig. 3.1 would be “3” and “9.” Humans demonstrate the ability to appreciate quantity when they are infants.13 The experimental protocol typically used to study numerosity in infants is surprise-them-and-they-look-longer. Infants are shown a small (subitizable) number of objects, which are then hidden before reappearing in either the same or a different quantity. When a different-than-expected quantity appears, infants tend to look longer than they do when the same-as-expected quantity returns.14 Looking longer is interpreted as surprise or interest, the salient point being that infants react when the correct number of objects does not reappear because they can identify and distinguish the small quantities involved. The young age at which infants show the ability to appreciate quantity – just months old – suggests that numerosity is both innate (biologically endowed; present at birth) and inherently alinguistic (independent of language) because it is found in human infants before they can speak or have had much exposure to language. Elicitation techniques with animals differ because, like human infants, they cannot respond to questions in language. The experimental protocol often takes the form of giving animals a choice between larger and smaller quantities of something and then seeing how they react. Fish, given a choice of two groups of other fish to join, reliably choose the larger group, a preference for what might be called safety in numbers that also demonstrates the ability to discern the difference in quantity.15 Salamanders, given a choice between more flies to eat and fewer, reliably choose more.16 These experiments show that animals can appreciate quantity, and moreover, that their sense of quantity functions similarly to the way ours does.

13

14 16

Starkey et al., 1990; Brannon & Roitman, 2003; Xu et al., 2005; Izard et al., 2009; Coubart et al., 2014. 15 Carey, 2009. Hager & Helfman, 1991; Dadda et al., 2009; Piffer et al., 2012. Uller et al., 2003; Krusche et al., 2010.

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     Humans share numerosity with a wide variety of other species: mammals, birds, reptiles, amphibians, fish, and perhaps even insects. Despite the vast differences of brain form and function that characterize the animal kingdom, the wide phylogenetic distribution suggests that the ability to discern quantity is evolutionarily advantageous, enabling individuals to choose more for things like food and shelter and less for competitors and predators.17 The phylogenetic distribution also means that the species ancestral to humans not only had the ability to perceive quantity, they had it well before language emerged. To aid the following comparison, we will limit our definition of numerosity to only what primates have, so we are sure to be comparing brains with similar form and function. Because numerosity is shared across the Order of Primates, it would have been present in the earliest hominoids, the group of primates that includes the gibbons, the great apes, and humans. Primates emerged within the class of mammals around 60 million years ago, hominoids 23 million years ago, and the last common ancestor of chimpanzees and humans some 7–13 million years ago. No theory of the evolution of language construes language as emerging as early as 7 million years ago, let alone at 23 or 60 million years. In fact, some estimates have language emerging quite recently, perhaps as little as 100,000 years ago.18 Thus, numerosity is evolutionarily older than both language and the human species. Numerosity is a key component of the nativist view because the innate capacity for numbers – the ability to appreciate quantity plus other innate capacities for things like numerical conceptualization and calculation – is deemed sufficient for numeracy. Numerosity (or numerosity-plus) thus provides a basis for finding, for example, concepts like zero in honeybees19 and rational numbers in chimpanzees.20 This view excludes a role for language, which is unique to humans, as well as material culture to the degree that only humans

17 19

18 Ansari, 2008; Coolidge & Overmann, 2012. Bolhuis et al., 2014. 20 Howard et al., 2018. Clarke & Beck, 2021.

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     have. The nativist view also implies that animals intuitively, spontaneously, and individually develop such concepts from interacting with the natural environment.21 In comparison, humans realize concepts like zero and rational numbers only after thousands of years of manipulating numerical information by means of material forms, a process of accumulating, building upon, changing, and transmitting information between individuals and generations that demonstrates that we do not develop numbers intuitively, spontaneously, or individually, even if other species do so. If the nativists are correct in seeing numerosity (or numerosityplus) as sufficient for numeracy and individual honeybees as spontaneously developing concepts like zero, we should be asking why it takes us so much more time and effort to develop comparable human versions of such concepts. This is an important challenge to the nativist view, especially given that humans and animals have essentially the same capacity to appreciate quantity, and whatever other cognitive capacities might be involved would surely be more highly capable in humans. It seems unreasonable to conclude that the things that differentiate us from other species – language and material culture, in particular – are epiphenomenal with regard to numbers, or worse, actually slow us down.

   Categorization is the cognitive process that groups or differentiates objects according to the similarities or dissimilarities of their properties, relations, or functions (Table 3.1). There are different theories about how categories are formed. In classical categorization, objects are grouped by the similarities of their properties. In prototype theory, some members of a category are more central to a category than others (e.g., a chair is more central to furniture than a tansu). In exemplar theory, an example that typifies the category is found,

21

Overmann, 2021e.

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

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Table 3.1 Categorical judgments. Cross-species differences in categorizing suggest that only humans might form concepts of numbers in the way we understand them. Data from Christie and Gentner (2007) and Gentner and Colhoun (2010). Previously published in Overmann (2019b, Table 4.1, p. 54). Judgment

Definition

Example

Found in

Identity

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

A is the same as A; A is not the same as B. AA and BB both consist of two identical elements. aaaaaaaa and BBB both consist of identical elements; aaaaaaaa and ♥♦♠ are both small in size; BBB and ♥♦♠ are both trios.

Humans; many other animals

Relations Cross-dimensional relations

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

     stored in memory, and used to judge whether things are like the exemplar. All animals can categorize to some degree, but humans stand out in this regard.22 In categorizing, the more properties objects share and the fewer objects there are to compare, the easier categorizing becomes; conversely, the fewer properties shared or the more objects involved, the greater the difficulty. Many animals, including humans, can judge identity, or discern sameness or difference between single elements, usually two objects that share most of their properties. Unlike most animals, humans can also judge relations, or discern sameness and difference in multiple elements, often properties shared by similar objects. Nonhuman apes, which are the species most closely related to humans, can also judge relations, but they have to be specifically trained to do it; even with training, they do not make these judgments spontaneously or achieve the same levels of complexity that humans do.23 Cross-dimensional relations involves discerning sameness and difference in only some properties of multiple elements. This requires inhibiting nonessential but highly salient properties to focus on pertinent but less salient ones like quantity, and it appears to be something only humans can do. An inability to connect quantity across sets may be why – despite the similarities of their brains, their perceptual experience of quantity, and their hands with those of humans – chimpanzees do not count on their fingers. Why does our ability to categorize exceed that of even closely related species? An important reason is our ability to leverage materiality for cognitive purposes. Material forms help to anchor and stabilize concepts, giving them a continuity far beyond what brains can achieve on their own.24 Inhibiting obvious properties while attending to less salient ones means attending not just to wholes but also to parts, evoking ancestral concerns with tool form and function.

22 23 24

Christie & Gentner, 2007; Gentner & Colhoun, 2010. Langer, 1986; Thompson et al., 1997; Thompson & Oden, 2000. Fauconnier & Turner, 1998; Hutchins, 2005.

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     This is something we can discern in the archaeological record as change in the design features of stone tools25 and the emergence and increasing complexity of composite tools26 across a span of tool use going back more than three million years.27 Our capacity for complex categorization may be an effect of our long history of using tools, a consequence of our ability to subdivide and conjoin material forms. 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 that are categorized; the organizing principles used for classification; whether categories are based on properties, relations, or functions; and the number and boundaries of the categories formed.28 This suggests that the differences between humans and animals in categorizing lie in material engagement per se, not in specific forms of material culture. The independence of categorization and the form of material culture, in turn, suggests that expansion of the ability to categorize may have begun early in the hominin lineage – the category that includes all species of bipedal apes, extinct or extant – perhaps when they first began to use tools several million years ago. How categorization might work in numbers, and why animals are unlikely to have concepts of numbers in the same way we do, even though they share the ability to appreciate the quantity of small sets, is suggested by the logical types defined by Bertrand Russell (Table 3.2).29 When individual objects (Type 1) are perceived, their subitizable quantities are categorized as similar or dissimilar through judgments of identity, something both humans and animals can do. Judgments of identity enable the pairing of objects with like objects (Type 2), something humans and nonhuman primates can do that implies the ability to manipulate material forms in addition to categorizing. Comparing quantity across sets of objects (Type 3) requires 25 26 28 29

Gowlett, 2006; Overmann & Wynn, 2019a. 27 Carvalho et al., 2009; Ambrose, 2010. McPherron et al., 2010. Hunn & French, 1984; Sillitoe, 2002; Unsworth et al., 2005. Russell, 1910; Soames, 2003.

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     Table 3.2 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. Data from Russell (1910, 1920) and Soames (2003). Previously published in Overmann (2019b, Table 3.1, p. 33). Type

Entity

Description

Example

1

Objects

a pair of objects

2

Sets of Type 1 Sets of Type 2

Individuals, objects, or entities Cardinality of a set of objects Cardinality shared by sets, a number

Sets of Type 3

A set of Type 3 entities

3

4

twoness (cardinality of a pair; quantity as we perceive it) two (twoness shared by two or more pairs; a judgment of crossdimensional relation) the natural numbers (of which two is a member)

the ability to judge relations, something humans can do that even closely related nonhuman primates cannot, perhaps because it requires inhibiting salient but irrelevant qualities to focus on unobvious but relevant ones. For example, judging that a pair of cars and two fingers share twoness – the number two – requires inhibiting all the salient properties that fingers and cars do not share to discern what is not nearly as obvious, the quantity they do share. This difference in categorizing likely makes quantity shared between sets – Russell’s definition of a number – something only humans can judge. The categories formed by classifying objects by their properties are abstractions, and abstraction is thus both a process and its product. As process and product, abstraction forms concepts by generalizing, decontextualizing, synthesizing, and reifying.30 Generalization forms broader notions from particular examples and applies them to new domains and contexts. Decontextualization extracts content

30

Dreyfus, 1991; Sfard & Linchevski, 1994; Ferrari, 2003.

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     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.31 Abstraction is often conceived as a purely mental activity. This view tends to exclude the recognition that both process and product are at least as much behavioral and material as they are mental. For example, realizing a concept of number from quantities shared between sets and then extending this concept to new sets of objects (generalization) implies interactions with sets of material objects, something we will look at in the next chapter. Similarly, dissociating concepts from the objects and sets that originally gave rise to them (decontextualization) depends on behaviors with materials, like using numerical counters to count in a location different from that of the enumerated objects themselves. Concepts like counting, quantity, and one-more and opportunities to notice and explicate patterns and relations (reification) can emerge from material engagements like making notches or rearranging beads. Essentially, interactions with material forms provide opportunities for our brains to do two things they are very good at: recognizing patterns and forgetting unimportant details. The term abstract has connotations that need to be acknowledged. Within psychology, abstract is not just a cognitive process and its conceptual products because it has also been used, as Piaget and Lévy-Bruhl once did in characterizing how societies think, to connote something that has become refined, rarified, or more complex, in the process becoming not just different but superior. Similarly, the abstract numbers associated with Western societies have been 31

Gray & Tall, 1994.

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     defined as concepts that have become independent or purified of the objects they once enumerated, while the concrete numbers associated with non-Western societies remain dependent and connected to them.32 As applied to specific numerical traditions, the terms abstract and concrete are often directly linked to work by Piaget and Lévy-Bruhl.33 Nonetheless, as a concept’s content changes, the nature of that concept’s neural realization within the brain – however imperfectly understood by neuroscience – does not. This insight has a practical value in modeling how numbers originate and elaborate, as it suggests that conceptual change can viewed as a matter of content, structure, and organization, and not value, status, or nature. Psychologists who study numerical cognition tend not to focus on categorizing and abstracting. This may be an artifact of what they do study, which for many researchers is seeing how Western brains respond to numerical stimuli. Western brains are enculturated into the Western numerical tradition, a system of numbers that is highly elaborated and mediated by symbolic notations. These numbers differ from those of other cultural traditions in many respects. For example, Western numbers have acquired a lot of properties over their lifespan. They have also become distributed over the many material forms used for their representation and manipulation, and this means they have also become functionally independent of any and all of them. They have also become relational entities, related to each other in myriad ways that can be manipulated by means of operations, with these manipulations being able to be performed both materially and mentally. While these qualities are the product of categorization and abstraction across many thousands of years, it also means that these processes are more applicable to diachronic studies of numerical change. This makes them tangential to the kinds of synchronic data collected during experimental studies.

32 33

Thomson, 1846. E.g., Schmandt-Besserat, 1978, 1992a; Damerow, 1996a, 1996b, 2010; see criticism in Overmann, 2018b, 2021b.

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    

    The mental number line (MNL) is the internal representation of numbers as falling along a linear continuum. One test of the MNL involves placing numbers along a continuum. Adults enculturated into the Western tradition tend to envision numbers as linearly disposed, that is, as evenly spaced across a straight line. In comparison, adults from societies whose numbers are emerging tend to envision numbers as more logarithmically disposed, that is, as smaller or bigger, so their graph is more curved. The cultural systems being tested are distinguishable as Western, old, and highly elaborated or non-Western, emerging, and little elaborated. The result suggests that the placement of numbers is influenced by culture. The MNL can also be tested by means of what is called the spatial-numerical association of response codes (SNARC). The SNARC effect demonstrates a left–right preference for subitizable (small) and nonsubitizable (large) numbers. That is, people tend to respond faster when subitizable numbers are presented to the left visual field, nonsubitizable numbers to the right. Conversely, reaction times are slower when small numbers are presented to the right visual field, large to the left. Why SNARC reaction times vary is unclear. Some researchers believe that reaction time differences show the presence of an MNL. Since similar effects have also been elicited from chickens,34 some researchers conclude that there is an MNL and that it is innate. Other researchers believe the MNL to be cultural and acquired through visual exposure to linear forms like written numbers and numerical graphs; reading direction is another potential influence. Exposure to linear forms and linearized behaviors may predispose Western people toward conceiving numbers in a linear fashion. Researchers investigating non-Western cultures, who can use forms that are less linear for counting like sequences of body parts that involve the hands, arms,

34

E.g., Rugani et al., 2015.

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     shoulders, neck, and head, have not found similar evidence of an MNL.35 In humans, cultural exposure is thus a plausible explanation for numerical linearization, though it does not explain the findings with other species. The possibility that the MNL may result from exposure to material representations of number is also suggested by variability in how numbers are constructed in different cultural traditions. In the Western tradition, numbers are built from multiples of ten that proceed in a linear fashion: hundred, thousand, ten thousand, hundred thousand, million, and so on. Some flexibility of construction is possible, but this flexibility tends to take the form of exchanging units and decimal multipliers across a linear continuum. For example, one million two hundred thousand can also be expressed as one point two million or twelve hundred thousand, and all of these are considered well formed.36 In Ibo, a West African language, 600 can be also expressed in multiple well-formed ways – “as 50 in twelve places, as 60 in ten places, as 100 in 6 places.”37 These Ibo constructions do not suggest an exchange of units and decimal multipliers across a linear continuum.

     The neurological substrate for numerosity is the intraparietal sulcus, a shallow cortical groove located toward the top of the parietal lobe, one of the four major parts of the human brain. The involvement of the parietal lobe in numbers is consistent across primate species:38 From monkeys to great apes and humans, neurons of the intraparietal sulcus are specialized to respond to numerical quantities.39 In humans, the parietal lobe supports an astonishing variety of

35 37 38

39

36 Núñez, 2011; Pitt et al., 2018, 2021. Chrisomalis, 2020, pp. 147–148. Jeffreys, 1948, p. 48. Le Gros Clark et al., 1936; Lewis & Van Essen, 2000; Biro & Matsuzawa, 2001; Choi et al., 2006; Tomonaga, 2008; Varga et al., 2010. Nieder & Miller, 2003; Cantlon et al., 2006; Fias et al., 2007; Ansari, 2008; Diester & Nieder, 2008; Hubbard et al., 2009.

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    

. . Homo sapiens and Neandertal skulls. (Left) The cranium of H. sapiens (20,000-year-old specimen from Abri Pataud, France) is rounder because of the distinctive human parietal enlargement. (Right) The Neandertal cranium (60,000–40,000-year-old specimen from La Ferrassie, France) is longer front to back and has the characteristic occipital bun. These shapes are used to diagnose the species to which a particular specimen belonged (see Kaas, 2000; Bruner et al., 2003; Bruner, 2004, 2010; Azevedo et al., 2009). Adapted from Stringer and Galway-Witham (2017, Fig. 1, b and d).

functions, not just numerosity and numeracy, but also finger gnosia (the ability to “know” the fingers), tool use, motor planning, perceptual–motor coordination, spatial cognition, 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.40 Beyond its involvement in these multiple functions, the human parietal lobe is remarkable in other ways. Its evolutionary expansion is perhaps the single characteristic that best distinguishes the human brain among primates generally, including the most closely related species, the Neandertals (Fig. 3.3).41 We share with them a common ancestor, Homo erectus. Neandertals flourished during the Middle 40

41

Ansari et al., 2005; Cantlon et al., 2006; Hamilton & Grafton, 2006; Orban et al., 2006; Diester & Nieder, 2008; Hubbard et al., 2009; Koenigs et al., 2009; Bruner, 2010; Bruner et al., 2014. Bruner et al., 2003, 2011, 2014; Bruner, 2004; Bruner & Holloway, 2010.

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     Palaeolithic (300,000–42,000 years ago) but went extinct shortly after Homo sapiens entered Europe at the beginning of the Upper Palaeolithic (42,000 years ago). While hominin brains, including ours and those of Neandertals, became bigger overall in relation to body size, in our lineage, the parietal lobes became even bigger still in relation to the other lobes. About 300,000 years ago, this nonallometric expansion caused the shape of the human brain to become more globular, or rounder, than is typical for primate brains, including Neandertals, whose large brains remained more elongated. Today, the differences in shape are diagnostic in assigning specimens to species. 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 the ecological niche of our ancestors, like tool use.42 Two things emphasize the close evolutionary relationship between the parietal lobe and tool use, in particular, between the numerical functions of the intraparietal sulcus and manipulable material forms. First, regions in the human intraparietal sulcus are specialized for representing additional aspects of visual stimuli. This might allow for a much more “detailed analysis of the object to be manipulated along many dimensions such as size, 3D orientation, 2D and 3D shape, etc., providing very sophisticated control of manipulation.”43 Second, “within the parietal lobes, adjacent neural regions encode numerical magnitude and grasping-related information.”44 When objects are sized so that we might grasp them by hand, our sensitivity to their numerical magnitude improves, suggesting an evolutionary relation between material manipulation and human numeracy.45 Besides the intraparietal sulcus, numerical tasks recruit other parietal regions, including the angular gyrus and supramarginal gyrus,

42 43 45

Rehkämper et al., 1995; Orban et al., 2006; Orban & Caruana, 2014. 44 Orban et al., 2006, p. 2664. Ranzini et al., 2011, p. 1. Simon et al., 2002; Ranzini et al., 2011.

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     adjacent ridges on the surface of the parietal lobe that are active when mathematical tasks are performed.46 There is evidence of significant linkage between the intraparietal sulcus and angular gyrus, which has been implicated in the expression and manipulation of number concepts. For example, 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 also more likely to be good at mathematical tasks.47 Gerstmann syndrome48 links damage to the angular gyrus with acalculia, the inability to process numbers; finger agnosia, the inability to differentiate among one’s fingers or hands; and impaired finger-counting, the inability to count on the fingers. This linkage is likely “the result of human-specific neural specializations [as well as] deliberate mathematical instruction and practice that fosters long-term connections between these areas.”49 The angular gyrus has also been implicated in a number of functions related to numeracy, including mathematical competence, the retrieval of arithmetic facts from memory, and the manipulation of numbers in verbal form.50 The supramarginal gyrus, which is also active when mathematical tasks are performed,51 has been implicated in inner speech, the internal experience of coherent but subvocal language.52 Along with the functionality of the angular gyrus and gesture, which is discussed in the next chapter, inner speech might help us access numerical intuitions and put them into words. Together, the intraparietal sulcus, angular gyrus, and supramarginal gyrus support functions like the use of the fingers in counting and the verbal expression of numerical concepts as metaphors.53

46 47 49 50

51 52

53

Dehaene et al., 1999; Zamarian et al., 2009. 48 Marinthe et al., 2001; Gracia-Bafalluy & Noël, 2008. Roux et al., 2003. Zamarian et al., 2009; De Cruz, 2012, p. 216; also see Zamarian et al., 2009. Lakoff & Núñez, 2000; Dehaene et al., 2003; Ramachandran, 2004; Grabner et al., 2007, 2013; Grabner, Ansari, et al., 2009; Grabner, Ischebeck, et al., 2009. Dehaene et al., 1999; Zamarian et al., 2009. McGuire et al., 1996; Lurito et al., 2000; Owen et al., 2004; Ardila, 2011; Geva et al., 2011. Lakoff & Núñez, 2000.

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     The parietal lobe is also involved in ordinality, the ability to construct and attend to sequences. In numbers, ordinality is one of two fundamental properties, with the other being cardinality. Ordinality orders numbers by increasing size, though not necessarily with an awareness that the increment is fixed at one. Ordering by size or magnitude is the property that Russell called of “enormous importance” because it gives numbers “most of their mathematical properties.”54 Two lines of evidence suggest that ordinality has a neurological basis: imaging studies, which show which parts of the brain are active when performing certain tasks, and synesthesia, a phenomenon in which the senses are strangely interconnected. In imaging studies, the anterior portion of the intraparietal sulcus is active during tasks involving ordinal sequences, not just in numbers but in things like the letters of the alphabet and days of the week; this region may also be involved in learning ordinal sequences.55 This portion of the intraparietal sulcus is also collocated with the intraparietal region active during “detailed analysis of objects to be manipulated,”56 suggesting that ordinality may have evolved for complex manual tasks involving sequences.57 And in synesthesia, the perceptual qualities of one domain – for example, color or taste – show up as unexpected and recurrent qualities in other domains, often those involving ordinal sequences. So, a synesthete or person with synesthesia might perceive numbers in terms of colors: one is red, two is green, three is blue, and so on.58 Since synesthesia is thought to involve cross-connections between neural networks, the fact that it often involves ordinal sequences suggests that the brain has a dedicated neural network for ordinality. Dedicated neural networks develop over evolutionary spans of time, which suggests that using ordinal sequences – including numbers – could have a very long history in our lineage.59

54 57 59

Russell, 1920, p. 29. Wynn et al., 2013. Wynn et al., 2017.

55 58

56 Fias et al., 2007. Orban et al., 2006, p. 2664. Harrison, 2001; Hochel & Milán, 2008.

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    

    The cerebellum (or “little brain”) underlies the occipital and temporal lobes at the back of the brain (or cerebrum), and while only 10 percent of the brain’s volume, contains 80 percent of its neurons.60 Historically, cerebellar functions were viewed as relatively circumscribed, limited to motor learning, motor planning, and the execution and control of motor movements.61 In recent decades, the cerebellum has become understood as involved in the mental manipulation of concepts in a manner similar to the motor manipulation of physical objects,62 as well as in functions like working memory,63 cognitive modeling,64 the detection of patterns and sequences,65 the formulation of higher-order rules,66 and creativity.67 Noting the importance of these functions to mathematics, the psychologist Larry R. Vandervert has suggested that cerebellar functions can explain the origin of mathematics.68 Interestingly, the H. sapiens cerebellum was larger than that of Neandertals, as assessed by the ratio of cerebellar volume to that of the cerebrum.69 Cerebellar volume correlates with executive functions like attention, flexibility (selectively switching between tasks), and working memory.70 Such differences in brain region size and shape suggests that there were possible cognitive differences between the two human species. This is currently a significantly contentious and ongoing matter of debate between two groups of scholars. One group argues that differences in brain morphology have cognitive implications with interpretive value for archaeological analyses. The other group argues that there were no differences between the two species, including their cognition,71 despite the many well-documented 60 63 66 68 69 70 71

61 62 Herculano-Houzel, 2016. Holmes, 1939. Ito, 1993, 2008. 64 65 Hayter et al., 2007. Vandervert, 2003. Vandervert, 2017. 67 Balsters et al., 2013. Vandervert, 2016, 2018; Coolidge, 2021. Vandervert, 2017. Weaver, 2005, 2010; Bastir et al., 2011; Kochiyama et al., 2018. Kochiyama et al., 2018. E.g., Zilhão, 2007, 2012, 2013; d’Errico & Vanhaeren, 2012; Villa & Roebroeks, 2014; Breyl, 2021; Vaesen et al., 2021a, 2021b.

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     differences in the parietal lobes,72 cerebellum,73 occipital lobes,74 orbitofrontal cortex,75 temporal poles,76 and olfactory bulbs,77 as well as rates of brain growth and development.78 Neandertal tools were certainly not less complex than those of H. sapiens, as their Levallois technology argues against this. The Levallois technique involves carefully prepping a stone core in order to produce a large flake from it; the technique is complex, difficult to learn, and requires significant mastery to execute.79 Comparisons of tool forms and manufacturing techniques show the two human species were essentially indistinguishable in their technical expertise, the ability to execute complex procedures like Levallois; however, they differed in their creativity, the ability to innovate and improvise.80 Not only is this finding consistent with well-known cerebellar functions and well-established differences in respective cerebellar size, it may have explanatory value for phenomena in the archaeological record. The first is the virtual “explosion” of complex material forms that becomes apparent in the archaeological record of H. sapiens about 100,000 years ago; the second is the relative statis of Neandertal material culture over a significantly longer period of time.81 Even when the two species inhabited the same environment at the same time, H. sapiens activities and artifacts noticeably outnumber those of Neandertals (difference in quantity) and were significantly more complex (difference in degree);82 this suggests that the differences in quantity and degree are unlikely to have been caused by environmental context or artifactual preservation. The differences in cerebellar size and the involvement of cerebellar functions like creativity in mathematics, and in tool forms and functions more generally, also imply that Neandertals were less likely 72 73 75 78 80 81 82

Bastir et al., 2011; Gunz et al., 2012; Neubauer et al., 2018. 74 Weaver, 2005, 2010. Pearce & Dunbar, 2012; Pearce et al., 2013. 76 77 Bastir et al., 2008. Hublin et al., 2015. Bastir et al., 2011. 79 Gunz et al., 2010; Pinson et al., 2022. Van Peer, 1992; Schlanger, 1996. Wynn & Coolidge, 2004, 2010, 2012, 2019; Coolidge, Wynn, et al., 2023. Mithen, 1996; Wynn et al., 2016; Coolidge & Wynn, 2018; Coolidge, 2021. Wynn et al., 2016.

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     to have had numbers, despite recent suggestions in the literature that they did.83 This is especially true for numbers, whose elaboration depends on the incorporation of new material forms, a matter involving an element of creativity. The issue of cognitive similarity between the two human species will not be settled in the present work and is currently a matter of ongoing investigation and debate.

-    As noted earlier, there is significant linkage between the intraparietal sulcus, the neurological substrate for numerosity,84 and angular gyrus, the neurological substrate for finger gnosia85 that is also involved in mathematical competence, retrieval of arithmetic facts, and manipulation of verbal numbers.86 The cerebellum is involved in the planning and execution of the motor movements, including those of the fingers, and provides movement-related functions that are important to conceptualizing in domains like mathematics. Such interconnections and functions between the fingers and parts of the brain associated with numerosity, numeracy, and movement suggest that fingercounting is a neurologically predisposed behavior, which may explain its prevalence across cultural number systems and significant differences in numerical elaboration. As we will see in the next chapter, the visual experience of the hand as a material structure is also critical. Using the fingers (and toes) as a material structure for representing numbers influences them toward properties including discreteness, linearity, and grouping by tens, fives, and twenties. These properties are discussed in the next chapter as matters influenced by the use of the hand as a material device. Linearity is explored here as a matter

83 84

85 86

D’Errico et al., 2018. Nieder & Miller, 2003; Cantlon et al., 2006; Fias et al., 2007; Ansari, 2008; Diester & Nieder, 2008; Hubbard et al., 2009. Marinthe et al., 2001; Gracia-Bafalluy & Noël, 2008. Lakoff & Núñez, 2000; Dehaene et al., 2003; Ramachandran, 2004; Grabner et al., 2007, 2013; Grabner, Ansari, et al., 2009; Grabner, Ischebeck, et al., 2009.

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     also influenced by the physical organization of the neurological substrates for both the fingers and numbers. 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.”87 The hand, in fact, is both sensor 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.88 As might be expected from this quality, 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 dedicated to their motor control and sensory activity is nicely illustrated by the Penfield homunculus (Fig. 3.4), a tiny figure whose hands are larger than his head and body combined. The topographical layout of the sensorimotor cortices (Fig. 3.5) is a function of how bodies grow from fertilized ova: Cells are added in a head-to-tail sequence organized and governed by the DNA “blueprint.” The resultant linear organization is advantageous in allowing the motor and sensory domains to be represented “continuously and completely” in a coherent fashion, rather than becoming disconnected when it crosses anatomical boundaries.89 The physical organization of the neurological level also appears to influence how quantity

is

represented

in

higher-order

abstractions,90

in

a

linearized fashion. The neurological substrate for numerosity is also topographically structured: Lateral and medial regions of the posterior parietal lobe preferentially respond to high and low nonsymbolic quantities, respectively.91 While sense organs in general are linearly organized as a function of how bodies grow from single cells, no sense organ is

87 90

88 89 Malafouris, 2013, p. 60. Gallagher, 2013. Patel et al., 2014, p. 351. 91 Harvey et al., 2013. Harvey et al., 2013, p. 1124.

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    

. . The cortical homunculus. (Left) Penfield’s drawing characterized the topographical organization of the motor functions (as the linear sequence from genitals to swallowing) and the proportion of primary cortex dedicated to them (as the size of the body parts); a highly similar drawing, not shown, characterized the sensory functions (see Penfield & Rasmussen, 1950; Penfield & Jasper, 1954). The organization of the sensorimotor cortices has since confirmed by magnetic stimulation studies with conscious human subjects, as well as lesion studies correlating motor cortex injuries with impaired muscle control. Adapted from an image in the public domain. Previously published in Overmann (2019b, Fig. 4.2, p. 60). (Right) When the homunculus is drawn as a human image, the nonproportional amount of cortex dedicated to the motor control of the hands is readily apparent. Image © Trustees of the Natural History Museum, London and used with permission.

associated with numerosity, which spans at least three perceptual modalities: vision, hearing, and touch. The reason why numerosity is topographically organized is currently unclear, but its benefits are not. The brain’s topographic layout of quantity appreciation may interact with visually appreciated quantity stimuli, perhaps underlying the mental number line92 and/or predisposing us to order quantities by increasing magnitude. This functionality may overlap with the topographical mapping of object size, “potentially allowing consideration of both quantities together when making decisions”93 and

92

Harvey et al., 2013, p. 1126.

93

Harvey et al., 2015, p. 13525.

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    

. . Topographical layout of the motor and somatosensory cortices. (Far left) The insert shows the respective location of the motor and sensory cortices. (Left) The motor diagram. (Right) The sensory diagram; it is broadly similar but not identical to the motor diagram, 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 both. Adapted from images in the public domain. Previously published in Overmann (2019b, Fig. 4.3, p. 61).

supporting “the representation of higher-order abstract features in the association cortex.”94 Later chapters will examine in greater detail how matters like subitizing and categorizing affect the ways in which material devices are used. In the meantime, behaviors like finger-counting are the topic of the next chapter.

94

Harvey et al., 2013, p. 1123.

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

Bodies and Behaviors

Our perceptual experience of quantity means that without counting, we recognize quantities up to about three or four rapidly and unambiguously, and we appreciate quantities larger than this range as bigger or smaller in groups when differences are big enough to be noticeable. These ranges correspond exactly to the first numbers to emerge across cultures and languages, even those widely separated by distance and time: one, two, (maybe) three, (occasionally) four, and many, with many often further specified as big many and small many. In other words, the first numbers are consistent with the functions of numerosity, subitizing, and magnitude appreciation. Numbers, both subitizable and higher, are often thought of as emerging in verbal form. Although this view is consistent with the idea that numbers are primarily a linguistic phenomenon, it is not supported by the ethnographic data, which show that material forms like gesture and marks on the ground are used to express numbers in the absence of any vocabulary for them. While both spoken utterances and physical gestures are ephemeral, words persist as part of a language in a way that may be more accessible to cultural observers than gestures are, perhaps creating the false perception that language has primacy. What emerging numbers show quite consistently, whether in spoken or material form, is the use of the hand for their expression and representation. The question of whether the hand functions as a material device in numbers is contentious. Naysayers emphasize the hand’s biological or embodied nature and reduce gesture to a form of language.1 Like speech, gesture is ephemeral; nevertheless, however briefly the fingers instantiate quantity, what truly differentiates them from material forms like tally 1

Johnson & Everett, 2021.



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    marks is that they are made of living flesh, rather than wood or stone. This biological nature is pertinent, for the hand is closely integrated with the sense of quantity. While parietal functions like numerosity and finger gnosia are neurally interconnected, finger-counting critically depends on the visual experience of the hands, both those of others and one’s own, a quality that suggests it can be treated as a material device. Evidence for visual dependence comes from studies with people blind from birth, who do not count with their fingers,2 though they use their hands to sense and act and possess unimpaired language and parietal functions. This evidence suggests that without visual experience of the hand, fingercounting is not predestined by the hand’s biological nature or its neurological substrate. Additional support for this conclusion arises from the wide cross-cultural variability in finger-counting, which suggests that how the hand is used in counting is patterned not by biology or neurology, but rather, by its material substance and structure.

         Two examples of emerging numbers show the patterning of numerosity despite their wide separation by time and distance. The Andaman Islands lie in the Bay of Bengal, southeast 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 more than 25,000 kilometers over land. Yet both the Andamanese and Fuegians, at the time Western contact recorded their numbers, counted no higher than two or three and indicated higher quantities as many: [The] only numerals in the [Andamanese] 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,” [and] “innumerable.” (Man, 1885, p. 100)

2

Crollen et al., 2011.

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     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. (Martial, 1888, Vol. 1, p. 208, as translated)

Such accounts might create the impression that subitizable quantities are immediately conceived as numbers in the way suggested by Brouwer, but this does not appear to be the case. Ethnographic and linguistic data from Oceania and the Americas suggest that one and two emerge first, though not necessarily in that order.3 This circumstance implies the perceptual salience of subitizable quantities, the categorization of quantity similarity and dissimilarity in singles and pairs of material objects, and the abstraction of shared and different quantities as concepts of the numbers one and two. Three emerges later, not just as a consequence of its perceptual salience as a subitizable quantity but often as a combination of two and one (or one and two). A few behavioral descriptions show three as being constructed from the smaller subitizable numbers, like this one recorded in the late nineteenth century about the Bakaïrí, an indigenous people of northern Brazil:

If I put down 3 kernels and asked for the number, I never once got an answer from Paleko and Tumayaua [the Bakaïrí informants] or from others with whom I practiced less, without the pile being broken down into 2 and 1: The first pair of kernels was touched, often loosened to check, then fingers V and IV were touched on the left hand and “aháge” was said; then the single grain was touched, finger III was pushed to the left to join fingers IV and V, “tokále” was said and finally “aháge tokále” was announced. (Von den Steinen, 1894b, p. 408, as translated)

The Bakaïrí appeared to anchor their understanding of the quantity of the kernels with the quantity of their fingers. That is, they did 3

Eells, 1913a, 2013b; Lean, 1992; Closs, 1993.

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    not passively view the kernels, but instead actively manipulated them, and then used their fingers as a material device to help anchor and express the numbers involved. This suggests that numbers begin as the perceptible quantities of singles and pairs of objects, experienced visually and manually, and expressed materially. Expression often involves recreating the quantity in question with the body, as the Bakaïrí did, or pointing out a material form that exemplifies it, as we saw when the Abipónes expressed four with a term meaning “the toes of an emu.”4 The expression of quantity by means of gesture precedes its verbal expression precisely because a lack of preexisting words and explicit concepts inhibits only verbal expression, while gesture enables us to illustrate and then perhaps verbalize what is nascent and ineffable.5 Contrast the availability of gesture for quantity with its potential use in color, another perceptual domain that is cross-culturally patterned in its emergence by perceptual experience and underlying neurology.6 Both quantity and color are visually perceived through perceptual capacities that are standard across the human species. In both cases, words let us describe or verbally point to exemplars and refer to them in their absence. But we can also express quantity with our hands or impose marks on the immediate environment, as for example, by inscribing lines on a nearby surface. There is no parallel to this in color, since we cannot make our bodies turn red or blue or create red or blue whenever we want to express those colors. Essentially, when numbers are first emerging – when there are no preexisting words or concepts – numbers depend less on language than do other cognitive domains, the exact opposite of what linguists argue about the criticality of language to numbers, at least in terms of its emergence.7 Gesture is also used when communicating numbers between languages: Displaying the requisite number of fingers expresses

4 6 7

5 Dobrizhoffer, 1822, p. 169. Broaders et al., 2007; Goldin-Meadow, 2017. Berlin & Kay, 1969; Kay et al., 1991; Regier et al., 2005; Kay & Regier, 2006. E.g., Hurford, 1987, 2007.

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     quantity clearly and unambiguously. Both parties understand the quantity in question by seeing it, whether or not they are from a culture that has numbers, know the names for any numbers contained within their own language, or share a common language with which to express them.8 The missionary James Barker described the behavior for the Yanoama (given as Yanomami, Yanomamo, or Yanomama in other sources), a people of Amazonian Brazil:

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 [presumably, in language rather than gesture] is difficult for them to understand. (Barker, 1953, p. 487, as translated)

Using the fingers, and not infrequently, the toes, to express numbers and count things is typical, given the high percentage of number systems today that use numbers like five, ten, and twenty as their base.9 The religious scholar Rafael Karsten noted the relation between fingers, counting, and naming the numbers for the Jibaros and Aguarunas, 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”). (Karsten, 1935, p. 548)

8

Olderogge, 1982.

9

Comrie, 2013.

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    Similarly, the ethnologist Buell Quain observed the use of fingers and gesture by the Trumaí of the upper Xingú River area in central Brazil:

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 was Quain’s informant] wanted to indicate many beijú [fish cakes], he held up his hand to their height when stacked. (Murphy & Quain, 1955, p. 77)

Fingers might be used with spoken numbers, either for clarity or emphasis, or the fingers used alone, without words.10 In some cases, the meaning of the verbal expression might be unclear unless accompanied by a gesture.11 For example, for the Bashila people of the part of the Democratic Republic of the Congo once known as Zaire, the phrase for seven, tʃinɛ lubali, means four on one side, a phrase that does not easily convey the idea of the number in question; however, the phrase was accompanied by a gesture that displayed four fingers on one hand and three on the other.12 The fingers might also be accompanied by a verbal phrase that merely mentions and thus draws attention to the hand as it represents a number, either because the speaker does not know or have names for numbers, or prefers to avoid the long, cumbersome phrases that name them, or chooses not to name them when communicating with someone who speaks a different language.13 The Lanì of West Papua, Indonesia count on their fingers without specific names for the numbers in question, and fingers may even be preferred to words, something noted for the Amazonian Tukano: [A Lanì] 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 10 13

11 12 Olderogge, 1982. Olderogge, 1982. Tempels, 1938. Lichtenstein, 1812; Hutter, 1902; Gabriel, 1921; Gore, 1926.

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     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.” (Larson, 2014) Ordinarily [Tukano] 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. (Da Silva, 1962, p. 257, as translated)

Terms for five and ten, when they exist in a language, “are usually based on ‘hand,’ and terms for 10 to 20 on ‘foot’; crosslinguistically, these are typical sources for these values.”14 Five and ten are generally the first nonsubitizable numbers to be realized and named, a function of both the neurological substrate and factors like the hands’ visual salience. Five and ten then become building blocks for the gap numbers, the numbers four and six through nine that complete the sequence between the subitizable quantities one through three and the nonsubitizable quantities five and ten. The gap numbers are normally named by adding or subtracting subitizable quantities from the quantities instantiated by the hands, as for example, realizing four as two and two (two fingers on each of the two hands) or one from five (all the fingers less the thumb), six as five and one (all the fingers on one hand plus three on the other) or three and three (three fingers on each of the two hands), and eight and nine as five and two and five and three (all the fingers on one hand plus two or three fingers on the other) or ten minus two and ten minus one, respectively (the fingers of both hands with one or two fingers either folded or extended). This makes the fingers the typical basis of a number system: The ten fingers are the basis of decimal numbers; five fingers per hand and the twenty total digits of the hands and feet are the basis of quinary and vigesimal numbers, respectively.

14

Epps et al., 2012, p. 67.

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    The importance of gestural and material means of expression does not lie solely in their priority of use, for they also reveal something in addition to the perceptual experience of quantity: Gestural and material expressions also show the recognition that quantity is shared between sets of objects, the difference between the perception of quantity and the concept of number.15 The fingers act as the reference set by instantiating the quantity of the enumerated objects or target set; the gesture is understood as quantity because the fingers instantiate quantity. Verbal expressions are similar in this regard. When the Abipónes used the toes of an emu to exemplify four,16 the Mundurukú the arms for two,17 and many cultures the hand for five, these terms point to the reference set by describing an exemplar that shares and thus exhibits the quantity of a target set.

         Counting with fingers and toes leaves distinctive traces in numbers: Involving the second hand in counting or recruiting the toes creates natural groupings by fives, tens, and twenties. Numerical vocabularies often reflect an embodied basis: Terms for numbers like ten typically mean things like both my hands finished or [all the digits on] the upper half [of the body];18 words for hand and man also mean five, ten, or twenty; nouns like digit mean both finger and number, not just in English but in widely separated languages like those of the Brazilian Aimoré and Hudson Bay Inuit;19 and verbs for counting also mean fingering, found in the Siberian languages Chukchi and Koryak.20 For example,

Fridtjof

Nansen,

the

nineteenth-century

Norwegian

explorer, scientist, and humanitarian awarded the Nobel Peace Prize in 1922, described the numbers of the Greenland Inuit as words or

15 18 20

16 17 Russell, 1920. Dobrizhoffer, 1822. Rooryck et al., 2017. 19 Dixon & Kroeber, 1907; Closs, 1993. Conant, 1896; Richardson, 1916. Antropova & Kuznetsova, 1956.

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     phrases based on the fingers and toes; compare these expressions to those of the Amazonian Desana in Table 4.3:

[The Greenland Inuit] count upon their fingers: One, atausek; two, mardluk; three, pingasut; four, sisamet; five, tatdlimat, the last having probably been the original word for the hand. When [an Inuit person] 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). (Nansen, 1893, pp. 194–195)

While five-fingered hands are the human norm and most societies and individuals use their fingers to count, societies with very few numbers are not typically described as counting on their fingers, and among the societies that do count on their fingers, few do so in exactly the same way.21 Some societies start with the left hand, others with the right. Some go across the hand sequentially; others use the fingers in nonsequential order. Some start with the thumb, others the little finger. Some bend the fingers, others extend, tap, or shake them. Some count the same pattern on both hands; some reverse the pattern on the second hand, or use a different pattern altogether. Many count to five before switching hands, but some count to four, six, twelve, or fourteen before switching. And while most use the fingers, this is not universal either, as others use the segments, joints, or tips, a few use the spaces between the fingers, some involve the base of the thumb or the feet, and some even move up the arm to the elbow, the shoulder, and various parts of the head before proceeding down the other side of

21

Overmann, 2014, 2021d.

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    the body to eventually arrive at the other hand. So, everyone uses the same device for the same purpose, but there is significant variability in how the device is used. How the hands are used as device has an effect on numerical structure, as counting to five on each hand produces one structure, omitting the thumb(s) or using the segments or joints yet others, adding the other hand or the feet produces even more, and counting that includes the arms and head between the two hands produces yet another.22 And these combinations do not exhaust the possibilities. Yet another method of counting with the hands was known for the Yuki, a people of Northern California, and the Aymara, a people of the Andean highlands. The Yuki method of counting, which is no longer understood because of the loss of knowledgeable individuals and the lack of detailed historical descriptions, 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.”23 This practice yielded a number system based on four and eight. The description by the anthropologist Alfred Kroeber admits the possibility that one hand might have held the sticks between the fingers while the other manipulated them, something that seems more reasonable than tying up both hands with interstitial sticks. Unfortunately, neither the descriptions available to us nor the Yuki names for the numbers produced by this method of counting are sufficiently detailed for us to be certain of the actual movements or method. Similarly, the Aymara have been described as counting both the fingers and the spaces between them.24 As five fingers with four intervening spaces make nine and Aymara spoken numbers are decimal, an external position is presumably also included so that counting reaches ten per hand. Unfortunately, these matters have not yet been described in detail in the literature. However neuroanatomically predisposed we may be to counting with the fingers, it is also a socially learned behavior. Simply, while 22

Overmann, 2021d.

23

Kroeber, 1925.

24

Bennett, 1949, p. 614.

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     different cultures count on their fingers differently, within any particular culture, people count on their fingers in the same way as each other because people in social groups imitate each other. Imitation synchronizes how we count, so that everyone in the social group counts on their fingers in essentially in the same way. This social aspect to finger-counting helps reinforce its practice and anchor its meaning within a particular society, while also highlighting the importance of vision in imparting the behavior. Counting to ourselves generally uses the fingers differently than displaying a number to someone else does. Using the fingers for display is known as finger-montring, from the French verb montrer, meaning to show or display. While finger-counting patterns tend to be more sequential, finger-montring patterns are more visually distinctive, which is consistent with their purpose, making the patterns displayed by the fingers easy to see and tell apart. Interestingly, the patterns we select for both finger-counting or finger-montring are not necessarily the ones that are the easiest to produce, given the biomechanical organization of our fingers and hands. While most finger patterns are fairly easily made, a few are relatively difficult to produce: for example, extending thumb and middle and ring fingers while bending the index and little finger.25 Typically, the ring and little fingers move together because they share common musculature. At least one of these difficult patterns – bending just the little finger while extending all the other fingers and thumb – is relatively common in modern finger-counting systems; it is often achieved by using the palm to hold the little finger in place or the free hand to bend it. This assistance allows the difficult pattern to be part of a sequence that moves across the hand, either bending the fingers one by one, starting with the little finger and ending with the thumb, or extending the fingers one by one, starting with the thumb and ending with the little finger. The inclusion of a pattern that is difficult to make suggests that the sequence it helped form was prioritized over 25

Lin et al., 2000.

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   

. . Oksapmin body-counting. The midpoint is unique, the body parts used for 15–22 reverse the order used for 6–13, and the fingers proceed from thumb to little finger for both 1–5 and 23–27. Redrawn from Saxe, 2012, Fig. 12, p. 46. Image © Cambridge University Press and used with permission.

ease of production. This is likely because sequential order is easier to keep track of mentally, just like doing things the same way every time is, a function of the finite resources of cognitive processes like attention and working memory. Not everyone stops with the fingers or continues with the toes. More than a dozen cultural number systems in Papua New Guinea, like the one shown for the Oksapmin in Fig. 4.1 and Table 4.1, continue counting up the arm and head, then back down the opposite side of the body. These body-counting systems typically include an odd (unmirrored, unique) center point, which is typically the nose. The other positions are mirrored, more or less exactly, on the other side of the body and are differentiated linguistically by adding the phrase on the other side to the name of the body part, a phrase often shortened to other.26 The phrase “more or less” refers to the fact that while most of 26

Lean, 1992; Saxe, 2012; Owens, 2018.

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     Table 4.1 Oksapmin number-words. The completed cycle may also be indicated with the exclamation fu! Data from Saxe (2012) and Lawrence (2018). Right side  1, thumb (tiβəna)  2, index finger (tipneɾip)  3, middle finger (bumɾup)  4, ring finger (xətɾip)  5, little finger (xətxəta)  6, wrist (doβa)  7, forearm (beza)  8, inner elbow (kiɾ)  9, biceps (towat)  10, shoulder (kata)  11, neck (ɡweɾa)  12, ear (nata)  13, eye (kina)

Left side ([on the] other [side], tən-)             

27, other little finger (tən xətxəta) 26, other ring finger (tən xətɾip) 25, other middle finger (tən bumɾup) 24, other index finger (tən tipneɾip) 23, other thumb (tən tiβəna) 22, other wrist (tən doβa) 21, other forearm (tən beza) 20, other inner elbow (tən kiɾ) 19, other biceps (tən towat) 18, other shoulder (tən kata) 17, other neck (tən ɡweɾ) 16, other ear (tən nata) 15, other eye (tən kina )

Midpoint 14, nose (ɾuma)

the mirrored body parts reverse the order used for the original body parts (e.g., . . .ten, shoulder; eleven, neck; twelve, ear; thirteen, eye; [fourteen, nose, the unique center point]; fifteen, eye; sixteen, ear; seventeen, neck; eighteen, shoulder. . .), the fingers on both sides in the Oksapmin case follow the identical, unmirrored sequence: On both sides, the thumb is followed by the index, middle, ring, and little fingers. The use of the body extends counting to amounts higher than is typically achieved by using the fingers and toes. The example serves to emphasize that our biology and neurology do not predetermine how we count with the body, not just how we count on the hands but also including whether we limit counting to the digits at the ends of the limbs.

-  - The oldest linguistic traces of finger-counting come from protolanguages, hypothetical ancestral languages reconstructed from

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    Table 4.2 Proto-Indo-European numbers. The asterisks mark their reconstructed status; the superscripts, subscripts, parentheses, and diacritics reflect matters of pronunciation that are not important to the present discussion. Data from Sihler (1995), Blažek (1999), Justus (1999), Bomhard (2008), and Beekes (2011). Nr.

PIE numbers

Meaning

1 2 3 4 5

*hoi(h)nos *duoh1 *treies *kwetuor ¯ *penkwe

6 7 8 9 10

*(s)uéks *séptm *h3eḱteh3 *(h1)néun *déḱmt

100

*kmtóm

one two three four Suggests the word whole, implying a phrase like the whole hand. six seven eight nine May have meant two of something, quite likely the hands. Analyzable as ten tens.

words shared by their descendant languages. For example, numbers can be glimpsed through Proto-Indo-European, the reconstructed ancestor of languages found today in India, Iran, and Europe that is estimated to have been spoken between 6500 and 4500 years ago. Proto-Indo-European words for the numbers one through ten and hundred have been reconstructed from descendants like Hittite and Sanskrit. Most are short and etymologically opaque, unanalyzable terms whose form gives no hint to their origin (Table 4.2); they are similar in this regard to the words one through ten in English, one of Proto-Indo-European’s descendants. Others are etymologically transparent, analyzable as words suggesting the hands or compounds built from smaller numbers. For example, the word for five is analyzable as a term meaning whole, which suggests a shortened form of a longer phrase like the whole hand; similarly, the word for ten is analyzable

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     Table 4.3 Desana numbers. Data from Miller (1999) and Silva (2012). Nr.

Desana numbers

Meaning

1 2 3 4 5 6 7

yuhu pẽũɾ̃ẽahpikũɾ̃ã yuhuru mõhõtõ yuhuru mõhõtõ yuhuru nĩãɾã ̃ yuhuru mõhõtõ pẽɾũ nĩãɾã ̃

8

yuhuru mõhõtõũɾẽ̃ ɾũ̃ nĩãɾã ̃

9

yuhuru mõhõtõ ahpikũɾ̃ã nĩãɾ̃ã pẽmõhõtõ guburi peʔre-ri bãhã-

one (1) two (2) three (3) two pairs (2 + 2) one hand (5) one hand, one from other hand (5 + 1) one hand, two from other hand (5 + 2) one hand, three from other hand (5 + 3) one hand, four from other hand (5 + 4) two hands (5 + 5) one man (20)

10 20

as two of something, most reasonably the hands.27 Such numerical expressions and their anatomical roots are quite common across languages. Etymologically opaque or unanalyzable forms imply great age and/or frequent use, since long words and phrases that are etymologically transparent or analyzable tend to wear away through processes of linguistic change;28 such change occurs more rapidly for words that are used often. In emerging number systems, words for numbers up to ten are typically analyzable as lengthy phrases describing the different fingers of the hands; analyzability is thus characteristic of recent emergence, infrequent use, or both. The numbers of the Desana, a people of the northwestern Amazon who live in the border areas of Brazil and Colombia, are an example (Table 4.3). Most of the names are analyzable as phases related to the hand. Since the 27 28

Sihler, 1995; Blažek, 1999; Justus, 1999; Bomhard, 2008; Beekes, 2011. Epps et al., 2012, pp. 55–56.

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    numbers one through three are short and unanalyzable, they are likely older than the others. A preference for fingers over words seems reasonable when the names for numbers are so lengthy that their sequential recitation is cumbersome, relative to counting with short words for numbers. Consider counting from six to twenty with phrases whose meanings are along the lines of, “one hand, one from other hand; one hand, two from other hand; one hand, three from other hand;. . .” We are far from reaching twenty, and already it has been a lot to say. The sequence is also more difficult to keep track of mentally because lengthy phrases impose a greater demand on attention and working memory, relative to shorter words.29 The cumbersomeness of lengthy phrases that describe the various fingers motivate the retention and use of gesture as a means of expressing numbers, and gesture, in turn, supports the meaning of the phrases as they shorten and lose transparency. Cumbersomeness suggests that use, especially frequent use, creates the need for conciseness and convenience, qualities initially achieved by strategies such as codifying the expressions and then omitting words they have in common. Given enough time and frequency of use, phrases like all the fingers on one hand and two on the other will eventually wear away to short, unanalyzable words like seven. Ultimately, the short, unanalyzable forms become part of the mental lexicon, the long-term memory for the meanings and sounds of about 10,000 words;30 there they act as atoms, productive elements that enable the generation of compound names like twenty-seven by means of lexical rules.

      Why do all societies with more than a few numbers and most of the individuals in those societies count on their fingers? The ubiquity of finger-counting is partly a function of the neurological substrate for numbers, since the part of the brain that appreciates quantity (the 29

Bender & Beller, 2007.

30

Carreiras et al., 2015, p. 86.

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     intraparietal sulcus) and the part that “knows” the fingers (the angular gyrus) are interconnected, and if we lose the angular gyrus, we lose the abilities to finger-count and calculate.31 But even beyond this neurological substrate, the hands are usually within our visual field – not as things in the background of our experience, but as things we constantly use, consciously attend to, and visually guide. This is true not just of our own hands, but those of the people around us as well. We watch the hands of others for the gestures they make, an integral but nonlinguistic part of human communication and an implicit part of the observational learning and behavioral recreation that is foundational to the very fabric of human culture. So, hands are unusually salient features of our socio-cultural environments. In addition, everyone has the same hand, so everyone understands the quantity meant not just by the gestural display of the fingers but also by the phrase as many as the fingers on my hand. In sum, we count on our fingers because our hands are internally connected to the sense of quantity and integral to our abilities to express and communicate ideas without words, and also because as material devices they are convenient, accessible, common, salient, understood, standard, and – importantly – visually perceived. As noted previously, congenitally blind people do not count with their fingers. They do not see other people count with their fingers, and observation and imitation constitute both the mechanism for acquiring the behavior and the reason why people in the same society count on their hands in the same way.32 If this were all there was to it, and given the neurological connection between the intraparietal sulcus and angular gyrus, congenitally blind people might spontaneously develop their own, idiosyncratic, finger-counting, somewhat analogous to the way idiosyncratic sign language can develop among congenitally deaf people.33 Nonetheless, blind people are not known to develop idiosyncratic finger-counting. This suggests that the visual experience of one’s own hands, independent of any 31

Roux et al., 2003.

32

Volterra & Erting, 1994.

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33

Senghas & Coppola, 2001.

    embodied, internal experience of them, is a critical ingredient in finger-counting. Our predisposition to use the fingers influences our numbers, in more ways than grouping by fives, tens, and twenties. For example, using the hand in counting also makes numbers discrete. Remember that the upper limit of the subitizing range, about three or four, is approximate or “fuzzy,” and that the first nonsubitizable quantity to emerge is five, as expressed and represented by the hand. It is difficult for about three or four to remain fuzzy when the numbers one through five are represented on adjacent fingers; in fact, representing them on adjacent fingers makes them visually and thus conceptually discrete.34 Using the hand in counting is thought to influence numbers toward stable order, something in which the dedicated neural network for ordinality likely plays a part.35 Using the hand also makes numbers linearized. This is because finger-counting leverages the topographical organization of our sensorimotor cortices, a layout that is inherently linear as a function of the way an organism grows, from head to tail, from its DNA blueprint. Numbers also acquire linearity from the way the hand is used in counting. Not infrequently, finger-counting proceeds in a rough line across the hand, say, from thumb to little finger. Our visual interaction with the material properties of the hand, plus the linear organization of the neurological substrate, influence the expression of numbers with the fingers toward linearity, and this influences the associated concepts of numbers toward linearity as well. Cross-cultural variability shows that the way in which the hand is used for counting is not governed or predetermined by any or all of the factors that motivate its use. This leaves each society free to choose how to use the device. And if all the factors that make finger-counting ubiquitous mean that pretty much everyone in a society counts on their fingers and counts on their fingers the same way, they still do not necessarily explain why we use the same pattern 34

Overmann, 2018a.

35

Wynn et al., 2017.

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     every time. In actuality, there is a straightforward reason for this. As mentioned earlier, finger-counting is socially taught and learned. We count on our fingers the same way we see others do it. But no matter what the social pattern actually is, all known cultural variants choose some physical features of the hand to act as starting and ending points and then establish a routine to proceed between the two.36 We do this because routine and habit reduce the demands on cognitive processes like attention and working memory. Doing things the same way every time also increases the accuracy of the results. In other words, we use the hand for counting just we would any other material device that is not a connected part of the body. Before moving on to the role of motor movement in mental computation, there is one more interesting thing about the fingers that should be mentioned: Finger-counting spans the gamut of numerical elaboration, with the possible exception of systems in which numbers are just beginning to emerge. That is, Western people are just as likely to count on their hands as people whose cultural number systems are much less elaborated. This distribution highlights the way that material forms are used in numbers: As new forms are recruited, older forms are retained, rather than discarded. This is not uncommon, as newer and older technologies can often coexist, like computers are used alongside pencils and paper. It is also not uncommon for newer technologies to replace older ones, in the way that the ballpoint pen made fountain pens obsolete. Replacement does not seem to occur as often in numbers, as Western people not only continue to use their hands with their elaborated numbers, they also use tallies (卌), token systems like coins, and symbolic notations. Even if older forms become less used, their retention and concurrent usage are the reason why numbers become distributed over multiple forms, something that makes them independent of any particular form.

36

Overmann, 2018a, 2019b.

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    This independence is absent in number systems that count only on the hands or body.

    Moving the fingers may be essential to mental computation. Evidence for this comes from three sources. The first is the mental abacus, a technique for performing arithmetical operations by manipulating an imaginary device. Practitioners of the technique move their fingers to perform calculations just as if they slid actual beads along the rods of a physical abacus. In reality, they manipulate nothing but air as they quickly and accurately add and subtract long strings of multi-digit numbers, calculate the square roots of six-digit numbers, and multiply ten-digit numbers. Mental abacus discards language in favor of movement

and

memory,

especially

visual

working

memory.37

Interestingly, the physical movements of mental abacus do not need to be carried out physically or seen visually. Performance is unimpaired when practitioners keep their hands still or perform blindfolded.38 This suggests that planning the motor movements is critical, but executing and seeing them are not. The latter is significant in light of how important it is to be able to see the hand to acquire the behavior of finger-counting. Certainly, mastering the mental abacus requires visual input when learning and practicing the technique. The second source of evidence that moving the fingers is important to mental computation is gesture, communicative movement that is integrated with both numbers and language in interesting ways. Gesture makes our numerical and spatial intuitions accessible, to both ourselves and others, by making them visible. It also expresses, through iconic means, ideas that we cannot verbalize because we lack the explicit concepts, the necessary vocabulary, or both. Gesture adds meaning to what we can express in language, both through emphasizing and also by illustrating, diagrammatically and 37

Frank & Barner, 2012.

38

Brooks et al., 2014.

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     spatially, 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.39 The third source of evidence is the cerebellum, the “little brain” located at the top of the brain stem that has traditionally been understood as having a role in learning, sequencing, and controlling motor movements, especially fine movements needing greater control. The cerebellum is now also thought to support our abilities to recognize patterns, form and manipulate abstract concepts, and make higherorder decisions and rules.40 Emerging research shows that the cerebellum may also provide a “common computational language to movement and cognitive processes (including mathematics).”41 As movements and mental processes are repeated, the cerebellum creates efficient internal models that make both movements and mental processes “smoother, quicker, and progressively more error-free.”42 Our lineage has been using tools for more than 3 million years.43 The longevity of this interaction has an important implication for mental content: Our engagement of material forms with the hands 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 often function as if these components were present. These neural muscles44 may explain phenomena like daydreaming and imagination, as well as the participation of mobility-impaired individuals in cultural systems like numeracy. The interesting question for numbers is whether the involvement of movements reflects our evolutionary history for using tools generally, or counting technologies specifically. Perhaps it is a bit of both. In either case, tool use is something that our lineage

39

40 41 44

Goldin-Meadow et al., 2001, 2009; Broaders et al., 2007; Cook et al., 2008, 2010; Ping & Goldin-Meadow, 2010. Vandervert et al., 2007; Vandervert, 2009; Koziol et al., 2010; Balsters et al., 2013. 42 43 Vandervert, 2017, p. 4. Vandervert, 2017, p. 4. McPherron et al., 2010. Overmann & Wynn, 2019a, p. 10.

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    possesses to a far greater degree than is true of any other species, enough that it has changed the form and function of our brains. If the motor manipulation of material forms is critical to forming concepts of numbers in the way we understand them, then as we are the only species using tools to such an extent, we are likely the only species forming such concepts. This poses a significant challenge to nativist estimations of similarities between human and nonhuman numerical cognition.

      What do people do when their language has relatively few terms for numbers, perhaps only the subitizable ones and maybe terms based on the hands? Interestingly, they do not count with their fingers, even if they have names for numbers based on them. Instead, they use behaviors called pairing and one-to-one correspondence. In pairing, two objects are physically grouped, enabling an appreciation of their twoness. Pairing pairs, like the Bakaïrí did in comparing a pair of kernels and a pair of fingers, involves the recognition that pairs of any kind share twoness, and twoness shared by different pairs is the number two. Pairing can involve subitizable quantities, either oneness as the quantity of a single contrasted with the twoness of a pair, or one (or two) as the number shared by two singles (or two pairs). Larger (nonsubitizable) quantities of objects can be lined up or matched two by two, a repeated pairing or one-to-one correspondence that means that two sets of objects are assured of being equal in number. This behavior corresponds to something noted by Bertrand Russell: “In actual fact, it is simpler logically to find out whether two collections have the same number of terms than it is to define what that number is.”45 In one-to-one correspondence, objects in two sets are iteratively matched until none remain unmatched and none have been matched more than once. Putting sets into correspondence in this manner creates the opportunity to recognize that they share 45

Russell, 1920, p. 15.

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     cardinality. Where pairing involved small sets (singles or pairs), oneto-one correspondence can involve any number of items, including quantities that far exceed the ability of the language to name them. It is limited only by the ability to handle and keep track of the objects involved. One-to-one correspondence spans the range of numerical elaboration. For example, Western users might use tally marks (卌) while inventorying. Each tally mark (|) means one item, each item occasions one tally mark, and tally marks are not made without a corresponding item being enumerated. Western number-users might also tap their fingers while enumerating objects; each tap means one object. These repeated pairings represent one-to-one correspondence between the sets of enumerated items and the tally marks or finger taps that correspond to them. An Australian aboriginal with restricted numbers, in crossing the Outback, might paint a stripe of mud on his arm for each day of the journey, and on his return, erase one stripe per day, in order to keep track of the distance traveled (outgoing) and remaining (return).46 These pairings represent a one-to-one correspondence between the sets of stripes and days. Significantly, the latter example was documented for an aboriginal people whose highest number counted, some fifty years later, was much lower (two or three)47 than the number of stripes recorded (fourteen);48 the material form extended the capacity for keeping track of quantity without involving explicit concepts or names for numbers, in much the same way a modern rosary keeps track of prayers. Each bead means one prayer, and counting by means of explicit numbers is neither involved nor needed. Behaviors like pairing and one-to-one correspondence have been dismissed as mere compensation strategies – what peoples do to manage quantity when and because they do not have numbers. But this characterization entirely misses the point: Such behaviors are the

46 48

Morgan, 1852. Morgan, 1852.

47

Mathews, 1904, p. 732; also see Blake et al., 1998, p. 71.

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    very process by which number concepts are realized. They are material engagements, the moments where brains connect and interact with bodies and other material forms in ways that result in the recognition of new patterns and the generation of new concepts. We will see this again when we look at the use of material devices to represent and manipulate numbers in later chapters.

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

Language in Numbers

Since we are taking an archaeological perspective on numbers, our interest in language is relatively narrow. Language can provide insight into the use of the fingers as a material device for counting, as well as whether ancient peoples shared the perceptual experience of quantity that influences form and function in the material devices that follow the fingers. Language can also reveal something about the processes through which numbers emerge and become elaborated, particularly in the form of characteristics indicating that ancient numbers emerged and elaborated through the same processes observable today in contemporary peoples. Finally, language can provide a basis for estimating relative age in numbers, albeit with several caveats. It is probably best to begin by defining what language is: “communication based on symbols” that emerged from (and was thus preceded by) an environment of “socially constructed conventions,” material forms acting as icons, indexes, and symbols.1 Consistent with this evolutionary perspective and our narrow focus on numbers, language is viewed here as a means of accessing and expressing our numerical intuitions. In this regard, it is one of two such means, the other being the manuovisual engagement of material forms. Both means are interacting and mutually influential, while inherently distinct, independent, and contributing different things. This is not how nativists see language and material forms in numbers. Instead, they see nonhuman species as achieving numerical concepts and computations analogous to those of humans without having language or using material forms. This view does not explain a significant difference between human and nonhuman species: It takes 1

Barham & Everett, 2021, pp. 536–537.



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    tens of thousands of years for human societies to realize concepts like zero and rational numbers that are claimed to be innate in species like honeybees and chimpanzees.2 Nor does it explain the wide variability of numeracy observed across languages and cultures: Some societies have old and highly elaborated number systems, while others have emerging, less elaborated number systems or possibly no numbers at all. Linguists tend to agree with nativists that material forms are epiphenomenal, but see language as the mechanism whereby numerical concepts are either generated3 or brought into existence. The latter is the idea that concepts do not exist unless and until they assume linguistic form;4 this is sometimes framed as quantity becoming discrete only once it has been named in language, making naming the mechanism for realizing discreteness. This idea ignores the visual discreteness that is implicit to material representation. Like perceiving visual discreteness, perceiving quantity is not a linguistic process; nor are categorizing and abstracting the similarities and dissimilarities of quantities to form concepts of number, especially if other species can do this. Further, expressing those concepts initially depends less on language than do other perceptual domains. Verbal labels follow the manuovisual means by which numbers are originally expressed because gesture and material forms are available and can be used without preexisting numerical words or explicit concepts. Gesture in particular can bring forth meaning that might not initially be clear, even to the person making the gesture; perhaps gesture activates one or more of the motor-planning functions involved in conceptualizing numbers, thereby nonverbally expressing semantic meaning in a way that makes it available for later verbal expression. Since we are treating language as a capacity for accessing numerical intuitions, we will look at how it is influenced by cognitive 2 3

4

Howard et al., 2018; Clarke & Beck, 2021. Hurford, 1987, 2007; Chomsky, 1988, 2004a, 2004b; Hauser et al., 2002; Fitch et al., 2005. Wittgenstein, 1933.

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     processes like numerosity, memory, and recursion, as well as external factors like using the fingers and frequency of use. We will also look at the necessity and sufficiency of numerosity and language as claimed by the nativist and linguistic perspectives.

     Lexical numbers are the words we use for counting: one, two, three, . . . in English, a member of the Indo-European language family; nye, mbini, ntathu, . . . in Xhosa, a Bantu language; tata, tusha, turu, . . . in Thao, an Austronesian language; and anyent, atherr, irrpety, . . . in Alyawarr, a Pama-Nyungan language. Lexical numbers have two basic forms: atoms and compounds. Atoms are short, unanalyzable words that reveal nothing of their etymological roots. For example, the English words one through ten give no hint to their origin in finger-counting, except in their decimal organization. Less than a century ago, such unanalyzable forms were considered “proper” or “real” in a way that analyzable forms, often overt compounds related to finger-counting, were not;5 today unanalyzable and analyzable forms are understood to be differentiated by the age of the number system and concomitant exposure to processes of use, linguistic change, and memory. In contrast, compounds are longer, analyzable, words and phrases. Some, generally numbers up to ten or twenty and their first few multiples, show unambiguous etymological roots in structures like the fingers and/or toes: terms for five that mean [all the fingers on one] hand and terms for twenty that mean [all the fingers and toes of] a whole man or [counting all fingers and toes has been] finished. Other compounds, typically numbers higher than the first few multiples of ten or twenty, are generated by means of lexical rules, discussed below.6 Atoms, a category that includes the numbers one to ten and productive terms like hundred and thousand, are the result of a

5

6

Dobrizhoffer, 1822; Nansen, 1893; Von den Steinen, 1894a; Dixon & Kroeber, 1907; Karsten, 1935. Greenberg, 1978.

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    number system’s longevity, the frequency of use for number-words, and aspects of human memory.7 Atoms reside in the mental lexicon, the function of our long-term memory that stores the meanings and sounds of about 10,000 words,8 and possibly as many as 80,000 “if all the proper names of people and places and all the idiomatic expressions are also included.”9 Compounds, especially those for numbers higher than the first few multiples of ten or twenty, are not stored in the mental lexicon; they are instead generated by means of lexical rules. Rule-based production is neurologically associated with our procedural memory system, the function of long-term memory responsible for knowing how to do things like walk, talk, and use material forms for numbers; lexical rules involve the same brain regions and functions because they are “like skills in requiring the coordination of procedures in real time.”10 Putting atoms together to produce new lexical numbers might thus be considered analogous to a motor procedure that constructs them from physical elements. Lexical rules enable us to generate names for numbers that we do not already know and could not possibly learn or memorize. This makes sense, if we consider that filling the mental lexicon with the names for all possible numbers would have little benefit, since most are used very little. In essence, we do not say the number seven hundred and forty-nine often enough for it to be stored in the mental lexicon. Such storage might even be detrimental, since the names of all the numbers possible would undoubtedly exceed our capacity for memory fairly early in numerical elaboration, as well as crowd out all the other vocabulary stored in there. Instead of being stored and recalled, names for larger numbers are produced as needed by means of lexical rules. The atoms in seven hundred and forty-nine – four, seven, nine, and hundred – are stored in the mental lexicon, and lexical rules involving multiplication (seven times hundred; forty

7 10

8 Epps et al., 2012. Carreiras et al., 2015, p. 86. Ullman et al., 1997, p. 267.

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9

Aitchison, 2012, p. 8.

     times ten) and addition (nine plus forty; forty-nine plus seven hundred) act on them to produce the name. Storage in the mental lexicon subjects words to memory effects and processes of linguistic change. Words for the subitizable numbers are the most affected because they emerge first; this makes them the oldest number-words in any system and gives them the greatest exposure to these effects and processes. The critical point is this: Number-words do not emerge as atoms. Rather, they emerge as descriptions of material quantity and compounds composed of smaller quantities. Respectively, these are adjectival phrases describing qualities of material forms like by itself and with another (one and two); nominal descriptions of quantity exemplars like the toes of an emu and the fingers of the hand (respectively, four and five); and compounds meaning one and one (two), one and two or two and one (three), and two and two, three and one, one and three, or one from five (four). Because they are oldest, exposed the longest to memory effects and processes of linguistic change, and emerge when the number system has the least amount of structure, the subitizable numberwords are typically the most irregular terms for numbers in any language. They are also the number-words used the most frequently (Fig. 3.2). This too makes them likely to change. Frequent use makes them likely to erode and truncate from lengthy expressions to short, unanalyzable words, a process in which their verbal meaning is supported and supplemented by the use of gesture. Short words are more efficient and less demanding of cognitive resources like attention and working memory, so lengthy expressions are under pressure to shorten, especially when they are used frequently. Lengthy expressions can become shorter by omitting elements that are understood contextually and/or repeated. An example is the prefix tan- (other side) in Oksapmin body-counting, which can be omitted from the numbers fifteen through twenty-seven.11 This is because the higher 11

Saxe, 2012.

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    numbers both involve the other side as the body parts are physically counted off (i.e., physical representation) and follow the numbers one through fourteen as counted on the original side and midpoint (context). Once number-words become atoms, they are no longer under the same pressure to change. In fact, in the Indo-European, Bantu, Austronesian, and Pama-Nyungan language families, the short, unanalyzable words for the numbers one through five are unusually conserved, a quality that both shows the lack of pressure and suggests these words could be 100,000 years old or more.12 While this estimate of age does not establish when these “old atoms” might have emerged in the four language families in question, nor indeed establish that they are in fact that old, the idea that numbers emerged in the Palaeolithic is consistent with an evolved neural network for ordinal sequences, as well as archaeological evidence of material forms possibly used for counting. It is also reasonable to believe that these old atoms resulted from the same processes observable today. This is because there are enough demonstrable similarities between emerging and ancient number systems to make it more probable than not that both emerged through the same process, rather than modern numbers emerging one way and ancient numbers emerging in an entirely different manner.13 This means that old atoms are likely to have emerged as lengthy expressions that then shortened to atoms over time under conditions of longevity and/or frequent use. Adding this stage could make the numbers represented by old atoms much older than 100,000 years, again, assuming that the conservation estimate is accurate. There does not seem to be a way to estimate how long it takes for lengthy expressions to wear away to atoms. One reason for this lacuna is that such change reflects how often numbers are used, variability that is governed by socio-material factors. Simply, linguistic change occurs more rapidly when use is frequent, a condition associated with large, interconnected societies; conversely, it occurs 12

Pagel et al., 2013; Pagel & Meade, 2017; Calude, 2021.

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13

Overmann, 2023b.

     more slowly with infrequent use, associated with small, relatively isolated societies. Another unknown is the degree to which the conservation estimate takes into account the dedicated neural network for ordinal sequences, which in modern populations conserves ordinal sequences like the counting numbers and the letters of the alphabet. Being subserved by a dedicated neural network might unusually conserve small number-words once they achieve the form of atoms, the point at which they also lose the pressure to change; this in turn admits the possibility that the old atoms in the four language families may be much younger than their conservation rate suggests.

 , -,   The initial numbers to emerge across languages and cultures express the subitizable quantities one, two, (sometimes) three, and (occasionally) four. The first numbers to emerge beyond this range are commonly five and ten, typically as represented with gestures that display the fingers of one or both hands, phrases that mean as many as the fingers on my hand for five (or hands, for ten), or both. Using the hand influences how nonsubitizable numbers are conceived and named. For example, counting one through about three or four and five on the hand influences a fuzzy number like about three or four to individuate as three and four when it is represented by adjacent fingers whose material discreteness is visually appreciable. Similarly, using the hand to represent the number five means that the words for hand and five are often related, perhaps even identical. Stated more generally, words for numbers in the range one to ten have traits that are characteristic of numerosity and finger-counting. Since these traits are found across languages and cultures, they suggest that numbers emerge and become elaborated through common processes that involve the same resources. When these traits are found in ancient languages or protolanguages, they show that ancient peoples realized and elaborated their numbers in the same way observable today. For example,

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    Table 5.1 Sumerian and Akkadian words for the numbers one through ten. The meanings of analyzable terms are shown; all others are unanalyzable. Data from Powell (1971, 1979, 1987), Edzard (1980), Pettinato (1981a, 1981b), and Miller and Shipp (2014). Sumerian Nr.

Word

1 2 3 4 5 6 7 8 9 10

diš min eš limmu ya aš imin ussu ilimmu u

Akkadian

Meaning

Word

Meaning

išten ¯ šina¯ šalaša ¯ erbet(ti) ḫ amšat šeššet sebet(ti) samanat ¯ tiš¯ıt eš(e)ret

5+1 5+2 [5 + 3] 5+4

Proto-Indo-European has evidence of using the hand for counting in the analyzability of its terms for five and ten (Table 4.2). Here we will look at analyzability and finger-counting in the languages of Mesopotamia, the ancient civilization that also provides the world’s earliest archaeological evidence of unambiguous numbers.14 Table 5.1 shows the numbers one through ten in Sumerian and Akkadian, the Neolithic languages most closely associated with Mesopotamian numbers and writing. Though both are now extinct, Sumerian is a linguistic isolate, unrelated to any known language, while Akkadian is a Semitic language related to extant languages like Arabic, Aramaic, and Hebrew. In Sumerian, the numbers six, seven, and nine are compounds of five plus the appropriate subitizable number; the pronunciation of eight differs,15 suggesting it was either formed differently or borrowed, both of which are common in contemporary languages. Such formations typically result from using the 14

Nissen et al., 1993; Overmann, 2019b.

15

Edzard, 1980.

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     hand in counting and naming the numbers after different combinations of the fingers. In Akkadian, the same terms are unanalyzable. However, Akkadian numbers are a decimal system, so while its terms are unanalyzable, its organization around the number ten suggests that the hand was used in counting. Besides showing evidence of finger-counting, analyzability has implications for relative age. As the Akkadian words one through ten are unanalyzable but their Sumerian counterparts are not, Akkadian is likely the older of the two number systems. The reasoning is as follows: The subitizable numbers emerge first as a function of perceptual salience, typically followed five and ten as a function of using the hand, and then four and six through nine to fill in the gaps in the numerical sequence. Subitizable number-words tend to be used the most, so between their earlier emergence and more frequent usage, words for one through three are likely to become unanalyzable before words for six through nine. In this case, both languages are unanalyzable in the subitizable range, so this does not tell us anything about age. The Sumerian words for six through nine are analyzable, while their Akkadian counterparts are not. This suggests that Sumerian numbers are the younger and Akkadian numbers the older of the two number systems. A couple of caveats are in order. An assessment of relative age cannot establish when either number system emerged or how old it might be. These matters would require an absolute date, something that is difficult to determine because language leaves no archaeological trace prior to the invention of writing. Relative age says merely that a number system is likely older or younger than one to which it is being compared, assuming their data reflect similar points in time. Analyzability also provides no insight into the rate at which phrases and compounds become unanalyzable. As noted earlier, such change is predicted to happen more quickly when numbers are used more frequently. This admits the possibility that an older number system could be highly conserved through infrequent use and thus have analyzable terms, while a younger number system

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    could have elaborated faster through frequent use and thus have unanalyzable terms. Akkadian’s ancestor, Proto-Semitic, a reconstructed Near Eastern language spoken around 3750 BCE, had terms for thousand and ten thousand, a considerable extent.16 At about the same time, Sumerian numbers appear to have reached into the tens of thousands, as archaeologically attested by numerical impressions on clay tablets. For example, an administrative tablet from the Uruk V period (3500–3350 BCE) contains a sign whose numerical value in the Sexagesimal System S is 36,000.17 Such higher numbers do not develop when numbers are first emerging or when they are little used. If one number system counted into the tens of thousands and the other into the hundreds, the one with the smaller extent would likely be younger. In this case, since the extent of counting is about the same in both, nothing new is suggested about their relative age. Nevertheless, counting into the tens of thousands means that both numerical traditions were well elaborated by the fourth millennium BCE, which suggests they were already considerably old by this time.

      Sumerian and Akkadian numerical language shows the same patterning that modern languages do. In modern languages, because this patterning is influenced by numerosity,18 the same patterning in ancient languages suggests that ancient peoples had the same ability to perceive quantity that we do. It is possible that the patterning is really a function of the cognitive resources underlying numerosity and language: attention, memory, and object tracking. For our purposes, it would not matter if this were the case, since either numerosity patterns numerical language, or attentional/memory/object-tracking resources pattern both numerosity and numerical language. In the

16 17 18

Lipínski, 2001; Kitchen et al., 2009; Miller & Shipp, 2014. MW 0188/038, provenance unknown, from a private collection (cdli P342530). Overmann, 2015; also see Franzon et al., 2019; Rinaldi & Marelli, 2020.

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     second case, we would still infer the same ability to perceive quantity on the grounds that it would be highly unlikely for the underlying resources to pattern numerical language, but not the ability to perceive quantity that is so widely shared between species and so constant in its functionality in our own species. The evidence for numerical patterning takes two forms. The first is frequency of use, which can be estimated from a text corpus, a resource that might contain, for example, all the words used in online English-language newspapers and periodicals within a certain range of years, as compiled and analyzed electronically. These data are used to calculate the frequency of particular words relative to all the others in the corpus. A graph of word-use frequency for the lexical numbers one through twenty in English, Chinese, and Arabic was shown in Fig. 3.2. Word-use frequency for the ordinal numbers first through eighth for the same three languages is shown in Fig. 5.1. Plotting data from Mesopotamian texts produces similar graphs, albeit with some differences and caveats, as will be described. As noted in Chapter 3, the lexical numbers used the most frequently are the subitizable ones. The usage frequency for Sumerian lexical numbers can be analyzed through the corpus of writing reflected by the textual record of Mesopotamia. We need to amend our technique somewhat because there are considerable differences between an original system of writing, like the one that emerged in Mesopotamia, and a modern one like our own. While our writing system has been refined over thousands of years to express languages like English with good fidelity, an original system of writing has an uncertain and highly variable fidelity to language. Importantly, early Mesopotamian texts were mostly numbers, with or without signs that specified what the numbers counted, which were generally commodities traded or donated to temples. These records do not accurately reflect how numbers would have been used in spoken or written language. Accordingly, we will focus instead on ordinal numbers, where the textual record of Mesopotamia has a greater fidelity to natural language.

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   

. . Frequency of use for the ordinal numbers first through eighth in English (top left), Chinese (top right), and Arabic (bottom left), selected because they represent different language families and are also associated with electronic corpora suitable for analysis. As is also true of lexical numbers (Fig. 3.2), the subitizable numbers are used the most frequently, and frequency decreases rapidly in a way that evokes the Weber–Fechner constant (bottom right). Data from Xiao and McEnery (2004), Davies and Gardner (2013), and Kilgarriff et al. (2014), and image by the author. A version was previously published in Overmann (2019b, Fig. 6.4, p. 104).

Ordinal numbers express order numerically: first, second, third,. . . . Ordinal concepts emerge before lexical numbers do as words meaning things like before, next, and following; this accounts for the differences in form between one and first, two and second. Lexical numbers are soon co-opted to express ordinality – “soon” as in when only a few lexical numbers are available, not “soon” as in a short or specific amount of time – given the relation between three and third, four and fourth, and so on. Once lexical numbers are co-opted for

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     ordinal numbers, they influence the original nonnumerical expressions for first and second toward numerical meanings that are consistent with the rest of the sequence. These changes had already occurred in Mesopotamia, since ordinals were expressed in writing by adding a sign for a sound value to the appropriate number sign,19 analogous to 1st, 2nd, and 3rd in English and 第一, 第二, and 第三 in Mandarin Chinese. Mesopotamian ordinal numbers show the same frequency of use that lexical and ordinal numbers do in modern languages (Figs 3.2 and 5.1): high but descending usage for subitizable numbers, a steep drop-off at the upper end of the subitizing range, and low usage for nonsubitizable numbers. The correspondence between usage frequency and numerosity shows that either numerosity informs numerical expression in language, or both are governed by the same underlying cognitive resources. Like the lexical numbers one, two, and three, the ordinal numbers first, second, and third may be more salient as a function of their subitizability and relative social importance, like the gold, silver, and bronze contenders in an Olympic contest. In contrast, four or fourth and higher, though expressible, may be conceptually less distinguishable or important because the ability to distinguish quantity decreases significantly at the subitizing limit. Comparing Sumerian ordinal frequency to that of extant languages is complicated by more factors than just fidelity to language. An original system of writing undergoes dramatic change, especially in the first thousand years of its lifespan,20 while modern writing systems do not. Corpora for extant languages are also realized by extensive data mining, as facilitated by electronic availability and automated collecting and organizing; this produces compilations of words that range in size from hundreds of thousands to several million. In contrast, the Sumerian corpus is compiled from the relatively few texts that have been found, translated, published, and made available electronically 19

Jagersma, 2010.

20

Overmann, 2016a, 2021a.

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   

. . Sumerian ordinal frequency for the ordinals first through twentieth. (Left) The Old Akkadian period represents the low-fidelity state of the writing system; the sample is composed almost exclusively of administrative texts. (Right) The Old Babylonian period represents the high-fidelity state of the writing system; the sample includes a wide variety of textual genres. Both graphs show a roughly similar trend for ordinal frequency, with smaller ordinals used more than larger ones. Image by the author. A version was previously published in Overmann (2019b, Fig. 7.4, p. 126).

through platforms like the Cuneiform Digital Library, yielding a much smaller list of words with significantly fewer samples per item. There are also considerable differences in the timespans covered. Modern corpora cover days to years, while the Sumerian corpus spans hundreds to thousands of years, depending on how the different time periods are parsed. Document composition differs as well, with modern corpora emphasizing narrative texts like newspapers and journal articles and the Sumerian corpus, particularly in its earlier periods, including significant numbers of administrative texts analogous to grocery lists, receipts, inventories, and spreadsheets. To examine ordinal frequency across change in the writing system (Fig. 5.2), the corpus of the Old Akkadian period (2340–2200 BCE), the earliest time period with sufficient data for analysis, was compared to that of the Old Babylonian period (2000–1600 BCE),21 the

21

Overmann, 2016c.

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     time period in which the Mesopotamian systems of mathematics and literacy achieved their greatest heights.22 The Old Akkadian corpus is composed almost exclusively of administrative texts, while the Old Babylonian corpus contains a wide variety of genres. In both, the lowest ordinals had a higher frequency of use, higher ordinals a decreasing frequency of use. The distribution showed an unusual higher frequency for second over first, and significant frequency for the ordinals third to seventh. These features may reflect the relative paucity of texts in the corpora, the unsettled nature of writing, an influence of administrative texts, some combination of these factors, or other factors not identified here. Despite the differences between ancient and modern corpora, Sumerian ordinal frequency was comparable to that of English, Chinese, and Arabic (Fig. 5.2), suggesting the same influence of numerosity on usage frequency found in modern languages. A second line of evidence for the influence of numerosity on language comes from grammatical number, the feature that marks quantity in nouns23 and which is thought to help the acquisition of number concepts when present as a feature of language.24 Across languages, grammatical number is structured in a way that reflects our perceptual experience of quantity: one, two, sometimes three, and many, subdivided as big many and small many.25 English, for example, is a one-many system. Many Semitic languages have a dual form that marks two items, and several Austronesian languages have a trial form that marks three items.26 Across languages, grammatical number can be marked in eight different ways,27 and individual languages may employ several of them. English, for example, uses plural suffixing, adding -s, -es, or -ies to the end of nouns to mark their plurality: cat (the unmarked singular form, meaning one) and cats (the marked plural, meaning many). English also uses some internal

22 25 26

23 24 Robson, 2007a, 2007b. Corbett, 2000. Carey, 2009; Barner, 2012. Overmann, 2015; also see Franzon et al., 2019; Rinaldi & Marelli, 2020. 27 Corbett, 2000. Dryer, 2013.

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    vowel changes, as in man/men or mouse/mice; retains a few archaic forms like ox/oxen from Old English; and adopts occasional foreign conventions like criterion/criteria. Grammatical number, when it is present, can be modulated by animacy; where this is the case, nouns for animate beings (a category that includes people, animals, and deities) will be marked, while nouns for inanimate objects will not be. Grammatical number in English is insensitive to animacy, so it is obligatory to mark number for all nouns, regardless of their animacy status. In sum, grammatical number has three aspects: its structure, method(s) of marking, and whether and how it is modulated by animacy. Structure reveals the influence of numerosity or underlying cognitive resources informing both, while differences in marking and animacy suggest the independent development of this feature in the languages where it is found. Three of the ancient languages of Mesopotamia – Sumerian, Akkadian, and Elamite – had grammatical number but differed in their details of structure, marking, and animacy (Table 5.2). Sumerian had a one-many structure modulated by animacy, distinguishing singulars and plurals for animate nouns but not inanimate objects through plural suffixing, reduplication, and encliticization.28 Reduplication repeats a word to indicate its plurality: An English example, one marked with an asterisk because it is not well formed, would be *cat-cat for the plural of cat. Encliticization is the use of a clitic, a morpheme that acts like a word but cannot exist on its own;29 English lacks numerical clitics, though it has nonnumerical clitics (e.g., ’ll, as in she’ll, the contraction of she will).30 Akkadian had a onetwo-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.31 Like Sumerian, Elamite was a linguistic

28 30 31

29 Edzard, 2003; Rubio, 2005; Michalowski, 2008. Jagersma, 2010. Huddleston & Pullum, 2016, p. 91. Sayce, 1875; Goetze, 1946; Edzard, 2003; Huehnergard & Woods, 2008.

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     Table 5.2 Grammatical number in Sumerian, Akkadian, Elamite, and English. Similarities in structure shows the influence of numerosity. Differences in structure, marking, and animacy suggest that grammatical number developed independently in these languages, most likely before the Neolithic period of contact. Data from Sayce (1875), Goetze (1946), Grillot-Susini (1987), Khačikjan (1998), Edzard (2003, 2005), Stolper (2004), Huehnergard and Woods (2008), and Jagersma (2010). Language

Structure

Method(s) of marking

Animacy

Sumerian

One-many

Yes, animate nouns only

Akkadian Elamite

One-two-many One-many

English

One-many

Plural suffixing, reduplication, and encliticization Plural suffixing Plural suffixing and reduplication Plural suffixing

No Yes, animate nouns only No

isolate; it was not mentioned in conjunction with lexical numbers because its writing system gives no insight into their phonetic values.32 Elamite had a one-many structure modulated by animacy, and marked grammatical number with plural suffixing and reduplication.33 Grammatical number in all three ancient languages conform to the structure expected from the influence of numerosity (or underlying cognitive resources). Like the usage frequency did, this demonstrates that ancient populations had the same perceptual ability for quantity found in humans today. This finding is not unexpected: We would predict that ancient populations had the ability to perceive quantity because of the phylogenetic distribution of numerosity in species generally; we would also predict that ancient populations had the same ability to perceive quantity that we do because numerosity

32

Tavernier, 2020.

33

Grillot-Susini, 1987; Khačikjan, 1998; Stolper, 2004.

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    is common across the human species today, even across the WEIRD/ non-WEIRD divide. Nonetheless, confirmatory evidence is always good. This insight into ancient numeracy also helps to counter a long-entrenched view of early Mesopotamian numbers as being particularly rudimentary and concrete – that is, as having emerged and elaborated in a wholly different manner than numbers are observed to do today – based on interpretations influenced by the work of Piaget and Lévy-Bruhl.34 The differences in structure, marking, and animacy also imply that grammatical number developed independently in the three ancient languages. Independent development would have occurred before the languages came into significant contact with one another during the Neolithic, which was a period of intense cultural contact and co-influence. This means that grammatical number would have developed before the Neolithic – that is, during the Palaeolithic. Since grammatical number develops after lexical numbers do,35 lexical numbers would have emerged even earlier in the Palaeolithic. We had already concluded that numbers emerged in the Palaeolithic on other grounds, but again, confirmation is always best, especially through multiple converging lines of evidence. The commonalities between numerical language in ancient and modern languages also supports the likelihood that processes for emergence and elaboration are universal, even for populations widely separated by time and distance. Grammatical number can develop in conjunction with very few lexical numbers, as it is found in languages whose numbers count no higher than three or five.36 That is, while grammatical number can suggest that Mesopotamian lexical numbers emerged in the Palaeolithic, it does not imply that these numbers counted any higher than three or five. Elaboration beyond the first few numbers is

34

35

Schmandt-Besserat, 1981, 1982, 1992a; Damerow, 1988, 1996a, 1996b; see discussion in Overmann, 2019b, 2021b. 36 Overmann, 2015. Overmann, 2015.

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     motivated by change in socio-material complexity, something examined in the next chapter.

    Before we examine the claims of necessity and sufficiency that underlie the nativist and linguistic views of numbers, it is worth examining what lexical numbers reveal about numerical organization and structure, and how spoken forms interact with material forms. Across languages that count to hundred or higher, lexical numbers tend to be irregular up to the first few multiples of ten or twenty.37 (Chinese and other Asian languages are an interesting exception in this regard, as their spoken numbers tend to be as regular in this range as they are above it; the reason for this regularity is unknown, but may be an outcome of how material and linguistic forms interact.) Irregular formations suggest a realization process involving material forms, subsequent descriptive naming, and the eventual formation of atoms. Above the first few multiples of ten or twenty, lexical numbers tend to be constructed regularly, suggesting their production is rule-based and implying that atoms are available. Within any particular language, the point at which rule-based production begins is sometimes higher, sometimes lower; over time, it can become lower, suggesting that rule-based productivity expands to displace earlier, more idiosyncratic naming.38 Despite any irregularity in spoken forms, organization and structure are regular throughout, as can be seen in material forms like written numerals. These characteristics are typical across languages (with the exception of spoken numbers in Asian languages, as noted) and can be seen in Sumerian numbers (Fig. 5.3). Lower Sumerian lexical numbers show the use of material forms, their descriptive naming, and atoms. The words for six, seven, and nine are compounds of five plus the appropriate smaller number, forms that suggest finger-counting.39 The words for the multiples of ten up to fifty involve ten and twenty, amounts that suggest the hands 37

Greenberg, 1978.

38

Greenberg, 1978.

39

Edzard, 1980.

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   

. . Organization and structure of the Sumerian lexical numbers. Productive multiples of ten appear in bold, productive multiples of sixty in gray and bold. Asterisks mark lexical numbers whose phonetic values were unknown when the Assyriologist Marvin Powell Jr. conducted his landmark analysis. He assumed that these numbers were constructed and pronounced as written because they were omitted from the lexical lists, glossaries that contained otherwise exhaustive compendia of the signs used in writing Sumerian (see Powell, 1971, p. 46; also see Wagensonner, 2010; Veldhuis, 2014). Compiled from Powell (1971). Image by the author. Previously published in Overmann (2019b, Fig. 7.3, p. 117).

and feet. Productive terms higher than sixty described the material forms used to represent the amounts they designated. These were numerical counters made of clay, known as tokens: šar meant “ball,” and šar-gal meant “big ball,”40 terms consistent with the spherical

40

Powell, 1987, pp. 480–481.

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     tokens of different sizes used for these numbers. The words for one through five, ten, sixty, thirty-six hundred, and possibly 216,000 were atoms. Higher Sumerian lexical numbers had a regularity consistent with rule-based production, as the higher orders emerged at the predicted places: multiples of ten and exponents of sixty. Lexical rules produce names from atoms; as atoms take some amount of time to emerge in a number system – perhaps a considerable amount of time, albeit as influenced by frequency of use – rule-based production and the atoms they use imply that a number system is relatively old and that numbers are frequently used and thus socially important. As exemplified by Sumerian, rule-based production tends to emerge at a point higher than the first few multiples of ten or twenty41 and involves amounts that reflect material forms like fingers and tokens. This pattern – atoms and irregular compounds up to the first few multiples of ten or twenty, regularized terms thereafter – suggests that rule-based production is a secondary phenomenon, one that follows and emerges from an initial phase of naming that is based on material forms and whose parameters are set by them, presumably under the same conditions of frequent use that are also needed to realize atoms from lengthy phrases. Unfortunately, almost nothing is known about the way language might incorporate an external parameter like the ten-ness of the two hands for use in rule-based production. No one even seems to be looking for this, likely because everything important in numbers is assumed to occur inside the brain. In this neurocentric view, material forms are passive recipients of mental content, and the hand is part of the body, so material influences on productive grouping are either dismissed outright or considered integral to language as an embodied cognitive process. Since congenitally blind people do not count on their fingers,42 neither the use of the fingers for counting nor the parameters set by their amounts or manner of use appears to be internally determined or 41

Greenberg, 1978.

42

Crollen et al., 2011.

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    motivated. As mentioned earlier, this suggests the hand should be reclassified as an external material device. Once this is done, the hand’s physical features and manner of use as a device become important, early influences on numerical structure and organization. Seeing the hand in this way also admits the possibility that other material forms have similar influence. Which devices are used and how they are used in representing and manipulating numbers then becomes a major source of the variability found between cultural number systems, not just in their organization and structure, but also in their degree of elaboration. This is important because cross-cultural numerical variability cannot be explained by numerosity, categorizing and abstraction, neurological interconnections, the hand’s morphology, or counting strategies like pairing, as these are common to the human species, even across the WEIRD/non-WEIRD divide. Redrawing the boundaries of the cognitive system for numbers to include the hand and other material devices, as we do here, opens up new possibilities for investigation, including the ways in which the components interact and influence one another to produce organizational and structural outcomes. For example, the interaction between the material and linguistic components of numbers has implications for the stability of a number system and cross-cultural differences in acquiring and manipulating numbers. An example illustrating the former is Oksapmin body-counting, which uses the body as a material tally along with spoken numbers that name the various body parts (Fig. 4.1 and Table 4.1). Both the material and spoken forms are ordinal sequences. This makes their structure and organization congruent in a way that is less likely to provide contrasts with the potential to illuminate underlying principles, compared to material and spoken forms that are incongruent.43 Congruence between material and spoken forms is likely to influence a number system toward longterm stability, and indeed, body-counting systems are thought to have persisted over a long span of time. 43

Overmann, 2018a, 2019b.

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     An example illustrating effects on acquisition and performance is found by comparing Chinese and English numbers. In the material form of written numerals, the numerals one through hundred of both systems are perfectly regular in their morphology, organization, and structure (Table 5.3). Chinese spoken numbers are also perfectly regular in their morphology, organization, and structure. This is not true of English spoken numbers, which are regular in their organization and structure but irregular in their form: One to ten and hundred are atoms, eleven to nineteen are irregular, and twenty and higher are imperfectly regular. If English spoken numbers were formed the way Chinese spoken numbers are, eleven would be *one ten one, twenty would be *two ten, and so on. Regularity and congruence between the material and spoken forms have been proposed to underly the greater ease with which native speakers of Chinese acquire and manipulate numbers, relative to native speakers of English, whose spoken forms are irregular and thus incongruent with the material form.44 Three points are important to take away from this discussion. First, material forms influence how numbers are organized and structured, including in language. Second, material and linguistic forms of number can differ, and the interaction of their similarities and differences can influence a number system in matters like longevity, acquisition, and performance. Third, the differences between material and linguistic forms highlight their status as inherently distinct and independent means of accessing numerical intuitions.

    Nativists claim that numerosity, the ability to perceive quantity, is both necessary and sufficient for numeracy, the ability to reason with numbers. On the one hand, if numerosity were necessary for numeracy, then as all animal species have numerosity, all animal species should have numeracy. To date, no animal species has been found to lack the ability to perceive quantity, so there is currently no evidence 44

Cantlon & Brannon, 2007.

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Table 5.3 Chinese and English written numerals and spoken numbers. While both sets of written numerals are regular, Chinese spoken numbers are regular and congruent with the material form in a way that English spoken numbers are not because they include atoms like four, irregular constructions like eleven, fairly regular forms like twenty, and highly regular forms like sixty. Compiled from various sources. Chinese 一 二 三 四 五 六 七 八 九 十

yi èr san sì wu liù qi ba jiu shí

十一 十二 十三 十四 十五 十六 十七 十八 十九

shí shí shí shí shí shí shí shí shí

yi èr san sì wu liù qui ba jiu

English 二十 三十 四十 五十 六十 七十 八十 九十 一百

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

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

eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen

20 30 40 50 60 70 80 90 100

twenty thirty forty fifty sixty seventy eighty ninety hundred



     to disprove the claim that all animal species have it, and no way to test the hypothesis that numeracy can develop in its absence. On the other hand, while the comparative data can be interpreted as showing nonhuman species to have numeracy, the degree to which their concepts are true analogues of human numbers remains unsettled.45 For humans, the necessity claim is plausible, for numerosity patterns numeracy in everything from the initial expression of numbers to the way that even the most highly elaborated numbers are used, both in the form and function of the material devices used for numbers and in language.46 Sufficiency is a different matter. If numerosity were sufficient for numeracy, then as all human societies have numerosity, all human societies should have numeracy. All human societies not only have numerosity, they all have the same numerosity, regardless of whether they are WEIRD societies with highly elaborated numbers or non-WEIRD societies with very few numbers.47 But not all human societies have numbers, as the Pirahã are said to lack them.48 Further, not all human societies with numbers have them to the same degree of elaboration, as societies like the Mundurukú have very few.49 If numerosity were sufficient, then it should be able to explain this cross-cultural variability in numerical expression, but it cannot. Linguists claim language as both necessary and sufficient for numeracy. If language were necessary for numeracy, then as no nonhuman species has language, no nonhuman species should have numeracy. The linguist James Hurford stated the position succinctly: “Without language, no numeracy.”50 This claim may assume that all numerical thought is both consciously experienced and linguistic in form. However, much if not most thought is neither,51 and for most people, the conscious mental manipulation of numbers in verbal form

45 46 47 48 49 50

Nieder, 2017a, 2017b; Núñez, 2017a, 2017b; Overmann, 2021e. Overmann, 2015, 2019b. Gordon, 2004; Butterworth et al., 2008; Henrich et al., 2010. Everett, 2007, 2013; Frank et al., 2008. Pica, Lemer, Izard, & Dehaene, 2004; Rooryck, Saw, Tonda, & Pica, 2017. 51 Hurford, 1987, p. 305. Amalric & Dehaene, 2016, 2018; Morsella et al., 2016.

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    is a particularly inefficient way to calculate. The idea that language is necessary is also directly opposed to what nativists claim, as they find analogues of human number concepts and computational capacities in alinguistic species as diverse as chimpanzees52 and honeybees.53 As for sufficiency, if language were sufficient for numeracy, then as all human societies have language, all human societies should have numeracy. This is significantly challenged by the fact that all societies have language but a few societies have almost no numbers. As it did for numerosity in the nativist view, cross-cultural variability strongly signals that something other than language is involved in expressing whatever capacity of language might be responsible for numbers, if that were indeed their source. The Pirahã are often invoked in discussions of necessity and sufficiency because their language is said to lack both numbers and recursion. The latter, the ability to embed words, clauses, or phrases beside or within similar structures, has been hailed as the “only uniquely human component of the faculty of language”54 and credited as the mechanism of discrete infinity, the ability to generate nearinfinite novel combinations from finite sounds and meanings. These views have positioned recursion as the sine qua non for both language and numbers,55 a view that is highly problematic because it would classify any language lacking recursion as something other than language. Recursion is also found in mathematics, where it is often considered to be the mechanism whereby the successor function (n + 1) generates new numbers,56 though arguably, succession is an iteration, as it lacks several of the qualities thought key to recursion: self-reference, the ability to increase embedding depth, and a terminating condition.57 Recursion is implicit to the rule-based production of new numbers: Names for higher numbers like seventy-two are recursively

52 55 57

53 54 Clarke & Beck, 2021. Howard et al., 2018. Hauser et al., 2002, p. 1569. 56 Fitch et al., 2005; Parker, 2006. Odifreddi, 1992; Reuland, 2010. Karlsson, 2010; Taraban & Bandara, 2017.

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     generated from atoms like seven and ten that are stored in the mental lexicon.58 It is difficult to envision this generativity as the source of numerical concepts or their conceptual content; it would be like ignoring the perceptual system for color and the material forms that have color and then saying we have concepts of color because we can generate comparative and superlative forms for them: red, redder, reddest. Admittedly, this assessment goes too far in the other direction, for rule-based production lets us express numbers we have never said, heard, or thought of before, or seen in material form. The lack of recursion has also been proffered to explain Pirahã anumeracy.59 These claims are also contested because recursion is evident in Pirahã storytelling and demonstrated when Pirahã speak Portuguese, and the Pirahã language may include terms for one, two, and many.60 Recursive production is an important contribution to numeracy. This does not entail that its absence would prevent the subitizable quantities from being named, as they are available to perceptual experience. This is supported by the possibility that Pirahã may have terms for one and two, despite the possible lack of recursion. Initial terms for numbers are often descriptions of material forms instantiating appreciable quantities, suggesting that while recursive generativity may be important, so is the descriptive use of language, and so too are the material forms that instantiate quantities and the perceptual system that appreciates them. Similarly, if recursion were the mechanism generating names for the initial numbers – say, one through ten – naming would presumably continue beyond the subitizable range, rather than stopping, as it invariably does, at about three or four. Presumably, naming stops at the upper limit of the subitizing range because we name the quantities we can see, while we cannot and do not name quantities we cannot see, at least not until we have made them tangible and visually and conceptually explicit through the use of material forms. Recursion does not appear

58 60

59 Ott, 2009. Everett, 2005, 2007; Frank et al., 2008; Nevins et al., 2009. Gordon, 2004; Sakel, 2012.

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    to be a factor in this, but then, the comparative evidence suggests that numerosity has the same functionality regardless of whether or not a species has language. The comparative literature tends not to mention recursion, which suggests that it is difficult to test, not a significant factor, or both. This omission has more importance than marking merely the difference between species with and without language, for in humans, recursion is related to computation (as for example, the multiplication and addition needed to produce the spoken number seven hundred and forty-nine). Recursion may also reflect a capacity that underlies language and which is thus potentially pertinent to alinguistic species. Humans and their ancestral lineage exhibit recursion in behaviors like flintknapping and knotting,61 and the archaeological record shows that flintknapping emerged before language did (knotting might have too, but fibers do not preserve archaeologically like stone does). In both language and behaviors, the combinatorial extent and hierarchical complexity of human recursion far exceeds that of any other species. If recursion is a factor in both human and animal numeracy, extent and complexity would plausibly differentiate the two, a challenge to the nativist position. The later emergence of recursive production in conjunction with higher lexical numbers suggests that for humans, recursion is not a factor in initial naming, raising the questions of how and why it might emerge. Given the role of motor-planning functions in conceptualizing numbers and the neurological association of rule-based production with procedural memory, conditions might include the realization of higher numbers, the frequent use of material forms to represent and manipulate them, and the availability of atoms. If linguists are correct that language is necessary for numbers, then numbers, at least in verbal form, would have emerged after language did in the evolutionary history of our species. Nonetheless, numerosity is phylogenetically distributed in extant species in such a 61

Barceló-Coblijn & Gomila, 2012; Moore, 2020.

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     way that it would have been present in our lineage no matter how far back in time we look. As an example – and a fairly recent one at that – primates emerged some 60 million years ago, and they would have had numerosity at that point because all descendant primate species have it today. There is no argument that language emerged when primates did; rather, it emerged much later, somewhere in the lineage of bipedal apes ancestral to humans. When this occurred is uncertain. Some scholars have placed it as early as 1.8 million years ago based on anatomical changes suggestive of language that appear in the fossil record around this time.62 While this evidence is inconclusive, a significant behavioral change also emerged at 1.8 million years ago: retaining and reusing tools, rather than creating, using, and discarding them as needed. This behavioral change might imply, if not language per se, then the ability to form concepts related to material forms used as tools.63 Many scholars argue that language emerged gradually over the last 500,000 years, give or take, placing its development well after our lineage split from that of the Neandertals and leaving their linguistic capacity somewhat in doubt.64 A few scholars argue that language emerged suddenly, possibly through a genetic mechanism, and only very recently, at most 100,000 years ago.65 As equally uncertain as when language emerged are its form and the mechanism(s) through which it did: It may have emerged as prelanguage or protolanguage, a form more akin to animal communication and much simpler than full language, which is discretely infinite and syntactically complex.66 Alternatively, language might have emerged through gestural communication,67 two-dimensional representations like cave art drawings,68 grooming behaviors,69 or recursive, hierarchically combined motor movements like those involved in flintknapping and

62 63 66 68

E.g., Tobias, 1981; Holloway, 1983; Falk, 1987; Holloway et al., 2009. 64 65 Overmann & Wynn, 2019a. Botha, 2020. Bolhuis et al., 2014. 67 Bickerton, 2002; Botha, 2012; Tallerman, 2012. Hewes, 1973; Corballis, 1999. 69 Davidson & Noble, 1989. Dunbar, 1991, 1996, 2017.

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    knotting.70 As language leaves no archaeological traces before writing and few anatomical clues,71 researchers must infer these matters from indirect evidence.72 Efforts have yet to achieve a definitive answer. If nativists are right that language is not necessary and correct that nonhuman species possess the capacity for numerical concepts and computations analogous to those of humans, then this capacity would surely have been present in the human lineage well before language emerged, regardless of when, how, and why it did. In this case, numbers could be very old indeed. If this were the case, the concepts in question would be subject to the limitations construed for honeybees: Concepts would be intuitive, spontaneously realized by individuals, and incommunicable to others because of the absence of language and the relative poverty of material culture. Against the nativist view, nonhuman species, to the extent they form numerical concepts, may be limited to those that reflect the perceptual experience of quantity: If it is true that humans cannot go higher than the subitizing range without language,73 then this would surely be true for animal species as well. Our lineage has been using material forms as tools for over 3 million years,74 using them uniquely for about 2 million years, and communicating through gesture and material forms before language was available. Given the predominance of gestural and material forms of numbers in contemporary emerging systems, it is conceivable that gestural and material forms of numbers might have emerged in our lineage long before language did. If material culture is indeed the critical ingredient, then the fact that our lineage has used material forms in distinctive ways for the last 1.8 million years or so becomes important. This possibility must be tempered by the fact that the unique parietal encephalization associated with tool use does not appear in the fossil record of our species until roughly 300,000 years 70

71 74

Stout et al., 2008; Allot, 2012; Barceló-Coblijn & Gomila, 2012; Ruck, 2014; Putt, 2019; Moore, 2020. 72 73 Fitch, 2000, 2009. Botha, 2016, 2020. Hurford, 2007, pp. 91–92. McPherron et al., 2010.

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     ago, with behaviors that we recognize as things we do ourselves becoming unambiguous in the material record only around 100,000 years ago. If numbers require the kind of elaborated material culture increasingly attested over the last 100,000 years, it might be the case that only the human species has numbers, at least numbers in the way we conceive them. Unambiguous numbers do not emerge in the material record until very late, 5000 or 6000 years ago. All of this points us toward material forms, for even if numeracy manifests in other species without the use of material devices, it most assuredly does not do so in our species. In fact, if the only resource we had for computing was mental and the only form we had to manipulate thereby was verbal, we would not have gotten very far with even simple arithmetic, for our finite capacities of memory and attention would preclude the development of much complexity. In the upcoming chapters, we will examine the archaeological record using insights from the ways in which contemporary societies use material forms for numbers; these ethnographic data will guide not only what we look for, but how we interpret what we find.

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

Global and Regional Patterns

Scholars reconstruct the prehistoric population movements that ultimately distributed the human species around the planet from three sources of evidence: fossil specimens, archaeological remains, and DNA. While all three diverge in their details, they generally agree that an ancestral species, Homo erectus, migrated into Eurasia about 1.6 million years ago, and our own species, Homo sapiens, emerged in Africa around 300,000 years ago and had left it by 100,000 years ago.1 H. sapiens reached Australia by 65,000 years ago, Europe by 45,000 years ago, and the Americas by 15,000 years ago.2 The idea that today’s global distribution of material technology is related to these prehistoric population movements is not new. For example, it was the central thesis of the popular 1997 book Guns, Germs, and Steel by the geographer and historian Jared Diamond. While the idea has become generally untenable for several reasons, of particular concern are the “racist implications”3 of explaining the efforts of contemporary peoples as the result of diffusion, the spread of ideas from one cultural group to another, rather than as the fruits of their own inventiveness. An example related to numbers comes from the proposal that a base-60 system associated with the Kapauku people of Irian Jaya, the western half of the island of New Guinea, represented diffusion of sexagesimal numbers from Babylon.4 Avoiding the problem means examining number systems in their present-day contexts. This regionally and temporally focused approach considers number systems either as unique, standalone

1 3 4

2 Higham, 2021. Clarkson et al., 2017; Dillehay et al., 2017; Fewlass et al., 2020. Bowers, 1977, p. 322, critiquing Price and Pospisil (1966). Price & Pospisil, 1966.



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     systems that exist in particular environments with specific adaptations or as compared to neighboring systems whose adaptations are similar because of proximity, shared environmental context, and contact.5 This focused approach is consistent with the perspective adopted here that commonalities between number systems arise not from cultural diffusion from prehistoric sources, but rather, the universal tendencies and resources that underlie numeracy in the human species: cognitive capacities like numerosity, material structures like the five-fingered hand, and the socio-material conditions that motivate the explication and elaboration of numbers as a technology for managing complexity. The focused approach can incorporate the global pattern of migration and its effect on when numbers emerged across the planet by recognizing that at the time H. sapiens was migrating into Australia, Europe, and the Americas, there were at most a few numbers, and possibly none at all. This means that human societies have the same starting point – few numbers or none at all – regardless of their geographic location. Numerical elaboration is more difficult to explain because it does not correlate with environmental conditions or whether societies subsist through foraging, hunting and gathering, pastoralism, horticulture, or agriculture.6 Nonetheless, there is an unambiguous, if poorly understood, relation between numerical elaboration and whether a society is relatively small and isolated or large and connected. This relation positions numerical elaboration as the typical response to the need to manage the increased complexity associated with population size and interactions between social groups. Assuming, as we do here, that emergence and elaboration are panhuman processes, number systems that are currently emerging can provide insight into these processes, the method known as

5 6

Bowers, 1977. Stampe, 1976; Heine, 1997; Divale, 1999; Winter, 1999; Epps, 2006; Epps et al., 2012; Hammarström, 2015.

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     cultural analogy. Moreover, the insights should be generalizable to how numbers emerged in the past, as confirmed by the traces that contemporary number systems contain of their emergence and elaboration: for example, traces of finger-counting in numerical structure or vocabulary. Using the emerging number systems of contemporary peoples to understand number systems whose emergence can be glimpsed only through such traces does not position the emerging systems as prehistoric relics. Rather, it recognizes them as the creative products of living people whose pragmatic inventiveness and cognitive dexterity is admirable, informative, and worthy of serious study. Such comparative study also helps to contextualize Western numbers as one among many cultural systems, challenging the assumption that they represent what numbers are. With these things in mind, we can capitalize on the opportunities that emerging number systems offer for realizing new insights into the uniquely human aptitude for numbers.

  :      In discussing the relative age of Sumerian and Akkadian numbers, we noted that the subitizable numbers emerge first and are used the longest and the most frequently, so the terms for them in language become unanalyzable – etymologically obscured as to their roots – before those for any other numbers do. Here we will compare the global distribution of analyzability for the spoken numbers one through three with that of six through nine, with two key assumptions. First, we will assume that any differences in the frequency of usage and effects that this might have on the rate at which numbers elaborate will not obscure the global pattern: Analyzability or etymological transparency as an indication of relative age should stand out clearly as a central tendency in the data. Second, we will assume that five and ten can be excluded from the analysis, since they tend to remain analyzable as words and phrases associated with using the hands for counting long after the words for lower and higher numbers

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    

. . Geographic distribution of analyzable number-words (n = 4842, but only 452 shown). Analyzable low numbers (triangles, n = 124); high numbers (circles, n = 289); both (stars, n = 39); neither (n = 4390). The 4390 languages with no analyzable words are not shown; these are concentrated in Africa and Eurasia but were omitted because their inclusion obscures both the other data and the map itself. Number system data from Rosenfelder (2018); location data from the World Atlas of Language Structures. Image by the author. A version was previously published in Overmann (2019b, Fig. 6.2, p. 99).

become unanalyzable, and their inclusion would otherwise complicate or obscure the global pattern. To analyze the geographic distribution of analyzability for low numbers (one through three), high numbers (six through nine), both, or neither, a sample of nearly 5000 contemporary number systems was examined (Fig. 6.1). No analyzability was interpreted as indicating the oldest relative age. Analyzable low numbers, especially in the absence of any higher ones, along with analyzable terms for both low and high numbers, were interpreted as indicating the youngest relative age. Finally, analyzable high numbers with unanalyzable low numbers were interpreted as indicating an intermediate age. As predicted, number systems with no analyzability (n = 4390) were concentrated in Africa and Eurasia, the landmasses settled the longest; these were omitted from the figure because their inclusion obscured both the other categories and the map itself. Number systems with

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     analyzable terms for low numbers (n = 124) or both low and high numbers (n = 39) were concentrated in Australia and South America, the landmasses settled the most recently. Number systems with unanalyzable low numbers and analyzable high numbers (n = 289) were concentrated in Papua New Guinea and North America. This distribution fits the pattern expected if the number systems most likely to be emerging are those at the ends of the ancient migration arcs. Interestingly, several African number systems had analyzable terms in the higher range. Why number systems with a trait suggestive of relative recency might be found in the longest inhabited landmass is unknown. It is possible that they are as relatively young as their analyzability suggests, but as discussed in the last chapter, it is also possible that they are older systems whose analyzable terms have conserved. In addition to infrequent use, potential reasons might include the habitual use of the fingers instead of words to express the numbers six through nine.7 This practice has been observed in situations where gestures are more easily communicated than words through the absence of a common language. It is a matter warranting further study. It is worth emphasizing that the global pattern places the youngest number systems in the last continents reached at a time in prehistory when there were likely few numbers or none at all. The global pattern does not predict subsequent outcomes, which are instead governed by regional factors. This means that while the global pattern does not predict exceptions like number systems with analyzable terms in the longest-settled landmasses, it is also not invalidated by them. The global pattern also represents relative, not absolute, age. This means that it cannot provide insight into exactly when or where numbers initially emerged, or how and why they began to elaborate. For this, we will need to consider the archaeological evidence that provides insight into the first unambiguous numbers, the artifacts 7

Olderogge, 1982.

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     possibly used for counting in earlier prehistoric periods, the limitations of the available techniques for discerning the purposes and meanings of prehistoric marks, the socio-material conditions under which numbers elaborate, and how we can discern socio-material conditions in prehistoric societies. Proto-languages also provide some insight into numerical elaboration, generally limited to the past 10,000 years; this is discussed in conjunction with the regional pattern.

  :    -  While their starting points were determined by when people arrived, subsequent regional developments are unique (Table 6.1). Within any particular region, societies that are small and isolated tend to have few numbers, while societies that are large and connected tend to have elaborated numbers. This correlation is both well attested and Table 6.1 Regional comparison of transitions in subsistence (the transition to agriculture or *evidence of land management), sedentism (the earliest villages with, minimally, 300 residents and 3 hectares; see Bandy, 2008), and population (the formation of states). Data from Burrows et al. (2006), Bandy (2008), and Hood and Vermeule (2019); dates are approximations that are revised according to new discoveries. Region

Agriculture

Large villages

State formation

Australia Near East Europe China New Guinea Egypt South America Indus Valley Mesoamerica North America

30,000 years ago* 9750 BCE 8200 BCE 6300 BCE 6000 BCE 5200 BCE 4400 BCE 3800 BCE 2000 BCE 1 CE

– 8550 BCE 4300 BCE 5700 BCE – 4800 BCE 1550 BCE 3500 BCE 1300 BCE 700 CE

– 3700 BCE 3000 BCE 1900 BCE – 3100 BCE 300 BCE 2600 BCE 200 BCE –

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     not explained by subsistence patterns, food storage, trade, division of labor, or climate.8 However, numerical elaboration does correlate with socio-material complexity, which involves the amounts of possessions and resources needing management, the emergence of social requirements and systems of value motivating the enumeration and accounting of goods, and the invention and use of material technologies for counting and keeping records.9 These matters are consistent with larger groups and frequent contact, conditions that increase requirements for managing internal and external complexity. Sociomaterial complexity might be summed up as things to count and reasons to count them, or even more simply as wealth. Given that numerical elaboration does not correlate with subsistence patterns or other matters, it apparently does not matter what the things are, or what the society does to obtain or produce them, or even why the things might need to be counted, apart from the fact that they are valuable in some way. Once the neurocentric notion that numbers are a wholly mental phenomenon is relinquished, the reason why numbers elaborate – for example, increasing in their extent – becomes relatively straightforward. They are not a latent mental capacity that “switches on” in response to some (unspecified) external condition. Instead, they are the outcome of recruiting and using material devices for counting, which is the typical response to increased socio-material complexity. The need to manage complexity by counting things motivates the use of material devices for numbers, and the devices facilitate matters like counting to higher numbers. Once this perspective has been adopted, the questions that are left to be answered are also relatively straightforward. We no longer need to be concerned with finding “the” external condition that triggers an internal numerical response, but can instead focus on the use of devices: Why do we use material devices to

8

9

Stampe, 1976; Heine, 1997; Divale, 1999; Winter, 1999; Epps, 2006; Epps et al., 2012; Hammarström, 2015. Overmann, 2013a, 2013b.

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     represent and manipulate numbers? What conditions motivate the recruitment of new devices? What factors influence the types of devices selected? What do different device types offer in terms of exploitable properties? How do properties of the devices used become conceptual properties? And how is conceptual continuity maintained across different devices? As these questions are examined in detail in the next chapter, here we will focus on the regional pattern. In assessing the interaction of socio-material complexity and numerical elaboration, there are four categories to consider: Societies are small/isolated and numbers are few; societies are large/connected and numbers are elaborated; societies are small/isolated and numbers are elaborated; and societies are large/connected and numbers are few. A study conducted in 2013 considered numerical elaboration and socio-material complexity in societies from Africa, Asia, Australia, Europe, and North and South America (n = 33).10 The sample contained societies in three of the four categories. Societies that were small/isolated with few numbers or large/connected with elaborated numbers, along with the absence of any societies that were small/isolated with elaborated numbers, suggested that numbers elaborate when socio-material complexity is present and do not elaborate in its absence. The large/connected societies with few numbers suggested that socio-material complexity increases first, and societies respond by increasing their use of numbers, which then elaborate, for example, by increasing in extent through the use of material devices like tallies that are capable of representing numbers with a capacity and persistence greater than what the fingers provide. These results also highlighted the material aspect of human number systems: Beyond motivating numbers, material culture intensifies the tendency to resort to material solutions in solving problems; it also provides the material forms that can spark insight and innovation, numerical and otherwise.11 These matters too are discussed in the next chapter. 10

Overmann, 2013b.

11

Kirsh, 2014.

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    

. . The Upper Rio Negro cultural area, showing language families and numerical extent. The languages (n = 32) span five families: Tucanoan (n = 18), Arawak (n = 7), Naduhup (n = 4), Kakua-Nikak (n = 2), and Carib (n = 1). Most of the number systems are restricted (n = 23), a few count to twenty or higher (n = 6), while the remainder could not be characterized due to insufficient data (n = 3). In two cases, sources disagreed on numerical extent; for both, the higher range is shown. The data were drawn and the image adapted from an updated version of the map in Epps and Stenzel (2013) and used with permission. A version was previously published in Overmann et al. (2022, Fig. 1, p. 138 and Fig. 2, p. 145).

The Upper Rio Negro cultural area illustrates part of a regional pattern. The cultural area is a portion of the Amazon that overlaps or abuts the borders of Brazil, Columbia, and Venezuela (Fig. 6.2). It is home to a number of small-scale, traditional societies; historically, these societies were well isolated from the larger region by factors of terrain and distance. These societies tend to have few numbers; globally, they are positioned where numbers are predicted to be emerging. The number systems in the region are generally similar to one another, which suggests the use of common resources, including numerosity, the fingers, and language; the development of similar

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     socio-material adaptations to the same environment; genetic, linguistic, and cultural relatedness, as the same language family implies common descent; and geographic proximity, which makes contact and co-influence between neighbors likely.12 For example, most number systems in the Upper Rio Negro are unambiguously based on using the hands and/or feet in counting;13 the few that are highly restricted in their extent count to numbers in the subitizable range. All six languages that count higher than twenty are located toward the area’s western periphery, suggesting the likelihood of mutual influence. Despite all the reasons these number systems should be highly similar, they are not identical because different societies make different decisions based on their individual needs and priorities, particularly in regard to which material devices are used and how they are used. While the majority of these number systems are restricted in their extent, many contain analyzable terms that provide insight into the material devices used. The systems counting to five and ten or twenty use the hand, while the ones restricted to the subitizing range use distributed exemplars, features of the natural or cultural environment that typify a particular subitizable quantity. For example, terms for two include “eye” (used in the Hup language), since eyes reliably occur in pairs, and terms for three include “jar support” (used in the Yukuna language), likely a reference to “stones or clay pedestals placed in a fire in sets of three.”14 There is another potential reason for distinctiveness: Numbers and counting practices are shibboleths, ways of signaling group affiliation and identity and protecting cultural boundaries against outsiders and external influences. For example, while members of a society will count on their hands in the same way as each other, they are also likely to count on their hands differently or name the fingers

12 13 14

Overmann, 2013b. Dryer & Haspelmath, 2013; Epps & Stenzel, 2013; Chan, 2021. Epps et al., 2012, p. 67.

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     differently than their neighbors do, in order to maintain a distinctive cultural identity. The manner of counting, or the names used in counting, then identifies the person who counts in a particular fashion as belonging to that society or cultural group. This might be valuable in times of conflict, as for example, the periodic but continual warfare described as motivating the Desana to create a system of numerical signs used to pass information about enemy combatants.15 As interactions between societies with distinctive counting practices increase and involve numbers, differences will disappear in time through the adoption of communal terms and behaviors.16 Conversely, a high level of numerical distinctiveness among adjacent societies implies that numbers are not a significant factor in their interaction. Not too far from the Upper Rio Negro, several societies with elaborated number systems are found. At least one of the associated languages is now extinct and known primarily through Spanish records.17 Of these elaborated number systems, the one best known today is that of the Inka Empire, which had reached the height of its expansion and claimed most of the western coast when the Spanish first arrived in the early sixteenth century. The northern extent of the Inka Empire reached inland to within only about 500 or 600 miles from the Upper Rio Negro. The Inka number system counted into the tens of thousands, calculated on counting boards called yupana, and recorded numbers on devices made of knotted strings called khipu.18 As Spanish explorers penetrated further inland from the coast to the Amazon region, they reported khipus for the Warekena, who today live along the eastern periphery of the Upper Rio Negro.19 Here two caveats are in order: First, Amazonian khipus do not necessarily indicate diffusion from Inka sources, since the differences between the respective number systems suggests there was little-to-

15 17 18

19

16 Fernandes & Fernandes, 2006; Overmann et al., 2022. Saxe, 2012. Quilter et al., 2010. De Santo Thomas, 1560; de Acosta, 1590; de la Vega, 1609; Guamán Poma, 1615; Velasco, 1841. De Sampaio, 1825.

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     no contact or co-influence between them. Independent invention using resources available to both is also plausible, since knot records are a common technology20 and travel between the coast and continental interior remains challenging even today. Second, Amazonian khipus are not numerical: They are mnemonic devices used to recall (nonnumerical) events, rituals, songs, and chants.21 A few tallies were also reported in the cultural area, but these were unambiguously linked to later European contact.22 The devices used in the Inka and Amazonian number systems raise an important concern: The devices are either made of organic substances that tend not to survive for long in archaeological contexts, or they are such that they are never deposited there to begin with. Khipus, yupana, and tallies are made of plant and animal fibers and wood. These substances are highly perishable over long periods of time, though the amount of time can vary with the circumstances of deposition (organic materials that are less exposed can survive longer) and local climate (organic materials degrade more quickly under conditions of moisture and heat). Finger-counting and distributed exemplars referenced through gesture and words would not leave any archaeological traces. This matter of archaeological survivability has direct relevance to our attempt to understand prehistoric number systems. Prehistoric cultures made extensive use of organic materials, but our insight into them is greatly limited, if not outright precluded, by the perishability of such substances. This circumstance suggests that we are unlikely to find the majority of prehistoric devices used for recording and manipulating numbers, and with them, any sign of the numbers themselves – even if they were highly elaborated, let alone if they were emerging. Given that societies with highly elaborated numbers like the Inka may not use materials like bone and stone for recording their numbers, we must also ask whether and why prehistoric societies might use such materials for numerical recording.

20 22

Birket-Smith, 1966. Koch-Grünberg, 1921.

21

Chaumeil, 2005; Hugh-Jones, 2016.

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     The geographic proximity of the Inka and Amazonian number systems also raises some important issues: Since the categorical distribution shows that socio-material conditions change first and numerical elaboration follows, how much socio-material change occurs before numbers start to elaborate? When and under what conditions does counting into the hundreds, thousands, and tens of thousands emerge? Counting to these ranges likely appears long before states form (Table 6.1); for example, Proto-Quechua, the reconstructed ancestor of the main Inka language, includes terms for hundred and thousand in the Quechua I and II groups that formed between 400 and 500 CE,23 well before the Inka Empire would emerge some 700 or 800 years later in the early thirteenth century CE. Unfortunately, this cannot answer the question of whether such higher numbers would have emerged with large villages, the transition to agriculture, or earlier. Second, how do we discern these matters for Neolithic and Palaeolithic societies, especially for those living tens of thousands of years before the transition to agriculture? Socio-material conditions are difficult enough to approximate for contemporary societies, for reasons that include the incomparability of culture,24 the timelimited nature of ethnographic data,25 the difficulty of defining cultures in temporal and geographic terms, and the near-impossibility of crafting unbiased, comparable measures.26 Socio-material conditions are even more difficult to approximate for prehistoric societies, for additional reasons. First and obviously, the passage of time means that there is no opportunity to witness behaviors or question informants. Another reason is that the archaeological record tends to underestimate socio-material complexity because of the use and perishability of organic materials; this makes socio-material complexity increasingly difficult to estimate the further back in time we look. Third and critically,

23 26

socio-material

complexity

increases

first,

24 25 Adelaar, 2004, 2010. Smith, 1955. Currie, 2016. Kaplan & Manners, 1972; Pelto & Pelto, 1978; Ember & Ember, 2000.

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and

the

     elaboration of numbers follows. This means that if the former can be detected archaeologically, it does not establish that the latter was present, only an approximate “not earlier than” timeframe for its subsequent emergence. Thus, if we find evidence of large/connected societies in prehistory, it means only that their socio-material conditions were such that numbers were likely to elaborate – not that the elaboration of numbers had already occurred. An example of conditions likely to motivate the elaboration of numbers is found at Sunghir, a gravesite in Russia dated to roughly 30,000 years ago.27 Recent analyses show that the people buried there had low levels of genetic relatedness consistent with exogamy, marriage outside the group.28 This is the kind of thing we would expect for a society in frequent contact with other groups; isolated societies, in comparison, show the high genetic relatedness associated with limited mate choice and inbreeding. Three of the people buried at Sunghir were found with elaborate grave goods, including more than 13,000 beads of mammoth ivory that had been strung and worn as necklaces or perhaps affixed to clothing. Each bead has been estimated to require a minimum of forty-five minutes to produce,29 so the number of beads represented nearly 10,000 hours of labor. Beads in these quantities have implications for more than just social stratification and specialization: That much labor and accumulated wealth would surely have motivated the counting of beads, and hence, the elaboration of numbers, most likely into the tens of thousands. A historically recent society making beads in such amounts not only counted them, but used strings of beads as part of the material substrate for counting. The Pomo, a people of northern California, made small cylindrical beads from clamshells; when strung, these beads “served as a medium of exchange, a standard of value, and a means of storing wealth”30 known as wampum. The Pomo were observed to count beads into the tens of thousands;31 their motivation

27 29

Marom et al., 2012; Nalawade-Chavan et al., 2014. 30 White, 1999. McLendon & Lowy, 1978, p. 311.

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28

Sikora et al., 2017. Loeb, 1926.

31

     for counting beads “developed from the wealth they acquired by being the principal purveyors of the standard disk currency to north-central California.”32 The Pomo method of counting was described as follows: Each string had twenty beads, and in counting, four such strings would be replaced by a small stick that measured about 1.75 inches in length and was worth eighty beads; when five of these small sticks had been accumulated, they were replaced with one large stick that measured about 3.0 inches in length and was worth four hundred beads.33 Further,

When four hundred has been reached the counting goes on in units of four hundreds until ten of the larger sticks have been used and four thousand beads have been counted. Now another group of ten sticks is prepared. They are all equal in size, a little larger than the former bundle of ten, and have some mark to distinguish them. Each stick represents four thousand. It must not be forgotten that while you are going on with each of these counts the previous bundle must be counted before you can ‘put out a new stick’. Hence a great number of sticks are in use at one time. When each of these latter ten sticks have been counted, you reach the number forty thousand, xai-di-lema-xai. This is known as the ‘big twenty’. (Loeb, 1926, p. 230)

In addition to the sticks that represented strings of beads, the Pomo kept track of their wampum exchanges with other social groups by means of knotted strings. Each knot was equivalent to four hundred beads, and when completed, the knotted strings were “used as a check on the tying together of the counting sticks” accumulated during counting; the string and stick records were then “put away together in a bag and kept” until the wampum was ultimately disposed of.34 Importantly, the Pomo did not record wampum on materials like bone. 32 33

Loeb, 1926, p. 230, drawing on material from Kroeber, 1925, pp. 256–257. 34 Loeb, 1926, p. 230. Loeb, 1926, p. 231.

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     Beads first appear in the Pomo archaeological record between 2500 and 500 years ago, though these artifacts were made of stone and not shell; evidence of the clamshell beads used as wampum appears around the sixteenth century CE.35 This recent date is consistent with the analyzability of the terms for higher numbers, a quality that also suggests recency: The term for eighty, dol-a-xai, meant four sticks; the term for hundred, lema-xai, meant five sticks; the term for four hundred, kali-xai, meant one (big) stick; the term for four thousand, hadagal, meant ten (big sticks); and the term for forty thousand, xaidi-lema-xai, meant one hundred (big) sticks. While observational and linguistic evidence documents the Pomo’s highly elaborated numbers and method of counting, there is no archaeological evidence of either. And this is exactly the problem we face with the Upper Palaeolithic. We can find evidence of socio-material complexity – like the bead wealth and genetic evidence at Sunghir – suggesting, minimally, that the conditions for elaborating numbers were in place, and maximally, that counting into the tens of thousands had developed. We can find contemporary analogies, like the Pomo, which suggest that prehistoric counting into the tens of thousands was not unlikely. Without also finding materials like bone that were used for recording – and that contain not just marks but marks that are unambiguously numerical – we are left with questions and doubts regarding the numeracy that would have existed. As mentioned, the world’s first unambiguous numbers do not appear until relatively late in the material record, no more than 5000 or 6000 years ago. These are the Mesopotamian impressions in clay dated to about 3500 BCE,36 a period that falls between the earliest large villages and state formation in Mesopotamia (Table 6.1). This was a period of dramatic change in subsistence, sedentism, population size, and contact between groups – exactly the kind of socio-material conditions likely to motivate numerical elaboration. But by the time

35 36

Basgall, 1982; Stewart, 1985. Schmandt-Besserat, 1992a; Nissen et al., 1993; Overmann, 2016b, 2019b.

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     this number system becomes archaeologically visible, it was already counting into the tens of thousands. This suggests the likelihood that at least some of the numerical elaboration accumulated by 3500 BCE would have been realized during the Neolithic and Palaeolithic. In Mesopotamia, artifacts earlier than the clay impressions are suggestive but inconclusive. Small clay objects labeled “tokens” have been dated to as early as the ninth and tenth millennia BCE,37 but none have been found in unambiguous accounting contexts, and there are no reliable techniques for establishing that such artifacts were used for numbers.38 Inscribed bones dated to even earlier periods have been found in the Levant and Zagros Mountains,39 the areas from which the peoples who moved into Mesopotamia originated.40 While these artifacts look like possible tallies, they too have not been conclusively identified as having served a numerical purpose, again, through the uncertainty of the available diagnostic techniques. While these artifacts are suggestive of behaviors with material forms with the potential to occasion concepts like numbers,41 they are uninformative regarding when numbers might have emerged or elaborated. Finally, as discussed in the previous chapter, several characteristics of the languages most associated with numbers in Mesopotamia suggest that numbers would have emerged during the Palaeolithic; unfortunately, these are similarly unable to pinpoint when this would have occurred. Language provides some additional insight into the prehistory of numbers in the form of proto-languages. Again, the insight is limited because of the imperfect nature of the evidence. Proto-languages are

37 38 39

40

Schmandt-Besserat, 1992a; Moore & Tangye, 2000. Zimansky, 1993; Friberg, 1994; Englund, 1998b. Mallowan & Cruickshank Rose, 1935; Du Plat Taylor et al., 1950; Braidwood et al., 1960; Dayan, 1969; Merpert & Munchajev, 1971; Munchajev & Merpert, 1973; Redman, 1973; Davis, 1974; Tixier, 1974; Merpert et al., 1976; Copeland & Hours, 1977; Marshall, 1982; Kenyon & Holland, 1983; Voigt, 1983; von den Driesch & Boessneck, 1985; Watson & LeBlanc, 1990; Yalçinkaya et al., 1995; Coinman, 1996; Reese, 2002; Prévost et al., 2021; Prévost & Zaidner, 2022. 41 Lazaridis et al., 2016. Overmann, 2016d; Malafouris, 2021.

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     hypothetical languages reconstructed from vocabulary shared by descendent languages, including terms for numbers. Proto-languages generally illuminate an early state of language that is roughly contemporaneous with the Neolithic, though some can reach even further back in time. For example, Proto-Afro-Asiatic, which is ancestral to Proto-Semitic and contains terms for subitizable numbers, is estimated to have been spoken between 18,000 and 12,000 years ago.42 A reconstructed number term plausibly indicates that counting to that level was not only possible at the timeframe estimated for the proto-language, it was widespread and used frequently enough that today the term is shared across the vocabularies of multiple descendant languages. The converse is less certain, since the absence of a reconstructed term for a number does not exclude the possibility that counting reached to higher extents. But absence does imply that any higher counting that did exist was not widespread or frequent enough to have passed into later vocabularies. A modern analogy is again suggested by the Pomo: Where most Californian peoples counted into the hundreds,43 the Pomo counted into the tens of thousands. Absence might also mean that any languages with more elaborated numbers – including perhaps the language once spoken at Sunghir – simply did not leave any descendants. If writing had not developed in Mesopotamia, this would have been the fate of more recent linguistic isolates like Sumerian and Elamite and their highly elaborated number systems. Proto-Indo-European, which is estimated to reflect the state of language between 4500 BCE and 2500 BCE, contains a shared term for hundred,44 and a shared term for thousand does not emerge until the third millennium BCE.45 If we were to estimate ancient numeracy by this evidence alone, it would suggest that counting before the third

42 43 44

45

Blažek, n.d.; Ehret, 1995; Wilson-Wright, 2014; Huehnergard, 2019. Kroeber, 1925, p. 879. Sihler, 1995; Blažek, 1999; Justus, 1999; Anthony, 2007; Bomhard, 2008; Beekes, 2011. Bomhard, 2008.

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     millennium BCE was unlikely to have reached into the thousands – at least, that it did not do so often enough to have occasioned a term for it shared by descendant languages today. On the other hand, the archaeological record does provide unambiguous evidence of the kind of socio-material conditions known to motivate the elaboration of numbers, and at a considerable depth of time too, like the Sunghir burial. This suggests that we should not easily discard the possibility that Upper Palaeolithic peoples had elaborated number systems. As long as the few marks on prehistoric bones and stones are not unambiguously organized as numbers, interpreting these artifacts as numerical will remain a challenge. This is an area in which cultural analogies are invaluable.

  The global and regional patterns are important to establishing where and why contemporary number systems are currently emerging and elaborating. Two assumptions motivate this claim: First, contemporary emerging number systems have the potential to provide insights into the emergence and elaboration processes, and second, these insights are applicable to understanding how the emergence and elaboration processes would have worked in the past. That is, emergence and elaboration are assumed to be uniform processes that worked the same way in the past as they do today; we should be able to learn about how they worked by observing contemporary societies and then applying the insights gained to past societies. Using contemporary societies to interpret past ones is known as cultural analogy. Ideally, the ethnographic data used in such an endeavor would be collected using contemporary theories and methods. Unfortunately, the historical data were collected under conditions that most generously might be characterized as an absence of theories and methods but which are more accurately labeled as a compendium of bad practices. Indeed, the historical data are often problematic because they contain the kind of biases and misunderstandings that contemporary theories and methods seek to prevent.

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     Since the historical data are valuable as the best and often only insight into what number systems were like around the time of contact, their flaws are mitigated by their overall consistency, with each other and with contemporary observations. As previously explained, the idea is that while any particular historical observation may be questionable, its individual reliability improves when identical observations are made elsewhere under vastly different circumstances. Here a different issue related to the use of cultural analogy will be addressed, the claim that studying contemporary societies with emerging number systems has only “limited value” on the grounds that the “validity of ‘uniformitarist’ comparisons between past and present societies of hunter-gatherers has always been hotly debated.”46 It is true that the use of cultural analogy has a contentious and justly criticized history in archaeology.47 Using “the present of one society simply to interpret the past of another, especially as the present is often seen as a latter-day survival of [a] stage passed elsewhere in the world,” has even been called “immoral.”48 Nonetheless, the criticisms are truer at broader levels of culture than for narrowly focused domains like numbers. Further, conclusions need not adopt the nineteenth-century position that so-called primitive societies are “fossilised survivals from proto-historical or even [P]alaeolithic times.”49 Contemporary traditional societies are neither primitive nor fossilized survivals; they are vibrant, creative peoples with a unique capacity to teach us new things about the panhuman ability for numbers. It is also worth noting that the data considered for the present analysis were not limited to societies with traditional lifestyles, but rather, encompassed all societies with number systems, whatever those societies happened to be. Using the present to interpret the past is simply the idea that if past and present societies have some similarities, they are likely to have others, including those we cannot directly see in the

46 49

Schlaudt, 2020, p. 631. Leach, 1973, p. 763.

47

Ascher, 1961; Wylie, 2002.

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48

Gosden, 1999, p. 9.

     archaeological record. Analogies can be “unreliable because, if things and societies in the present and past are similar in some aspects, this does not necessarily mean they are similar in others.”50 Certainly, “an incredible variety of codes of behaviour in fact actuate human conduct.”51 Nevertheless, the more similarities there are, the more likely other traits will be shared as well, until “[at] some point, the volume and detail of comparable points [may] become so great that it is unreasonable to suppose differences in the few aspects for which comparability is unknown.”52 This is the fortunate situation we have with number systems. These are related in a myriad of ways that make them unusually comparable – from their starting point in species-universal resources (e.g., numerosity, physiology, behaviors, material forms) to outcomes of numerical content, organization, and structure that are strikingly cross-culturally similar despite significant differences of time, place, environmental context, material culture, social structure, and so on – to the extent that philosophers and mathematicians like Plato and Frege have tried for over two thousand years to explain the universal quality of numbers. Indeed, all the things that make number systems so similar are why number systems that are presently emerging and elaborating provide good insight into how the processes work. Given this level of similarity, it is nonparsimonious to conclude that insights gained by studying contemporary number systems would have little value or relevance for understanding past ones, a position implying that past number systems might have emerged in ways we do not find in contemporary societies, a nonparsimonious conclusion we should reject. The approach used here is very narrowly focused on numbers: numerical content, structure, and organization; aspects of numerical language, particularly in what they can reveal about material forms, behaviors, and psychological processing; the material devices and behaviors used to represent and manipulate numbers, including the 50

Hodder, 2012, p. 12.

51

Smith, 1955, p. 5.

52

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Hodder, 2012, p. 16.

     fingers and collaborative or sorting strategies; and the types of objects that numbers are used to count. This narrow focus does not compare societies, present or past, in toto: Finding similarities between number systems, associated behaviors like finger-counting, associated number terms in language, and material devices like tallies does not assert that societies are similar in other ways. Indeed, one of the challenges in understanding numerical emergence and elaboration is the inconvenient fact that these processes do not correlate with subsistence patterns, food storage, trade, division of labor, or climate,53 other domains in which societies might be compared. Socio-material complexity is a different matter. It is indeed difficult to compare societies in terms of their socio-material complexity because there are so many variables that might be considered and no commonly accepted methods for choosing among them or prioritizing their importance. Notably, the present work does not compare societies in terms of their socio-material complexity, as this was not needed to examine the role of material devices in numerical emergence and elaboration. Instead, the present work draws from studies that have performed this kind of analysis, as well as cultural descriptions of specific societies. For both, the literature used was recent and peer-reviewed, qualities that increase the likelihood of adherence to current scholarly and ethical research norms, relative to the historical material. If socio-material complexity is difficult to compare in contemporary societies, it is even more so for ancient ones, where the data needed are reduced or absent and the lack of methods and priorities still applies. In principle, the correlation means for ancient societies that small populations in relative isolation would have restricted numbers, while large, interconnected populations would have the conditions for numbers to elaborate, but not necessarily elaborated numbers, since these develop after the enabling conditions are

53

Stampe, 1976; Heine, 1997; Divale, 1999; Winter, 1999; Epps, 2006; Epps et al., 2012; Hammarström, 2015.

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     present. The challenge is assessing whether the enabling conditions are present, connecting the conditions to specific artifacts that might have represented numbers – assuming there are any – and interpreting the artifacts accordingly. As was the case for contemporary societies, the socio-material complexity of ancient societies was not assessed or compared in the present work, though cases with the socio-material conditions likely to have motivated the emergence and elaboration of numbers, like Sunghir, were noted. In general, any attempt to apply the correlation to ancient societies would need to do two things. The first is estimating how socio-materially complex a particular ancient society might have been, while the second is determining when, how much, and whether numerical elaboration might have developed in response. Both inquiries exceed the scope of the present work, but are worthy of future investigation. Currently, one of the biggest gaps in the literature involves cultural notational systems, including but not limited to those recording numbers. While this lacuna will be discussed later in the context of understanding Palaeolithic artifacts that may have been used for notational or numerical purposes, it will suffice here to note that cultural analogy is perhaps the best and only method for obtaining the insights needed for such interpretations. Unfortunately, the historical and even most contemporary literature tends to ignore notational devices or merely mentions them in an offhand way, rather than describing them to the levels of detail that would enable their use as comparisons for Palaeolithic artifacts. As the archaeologist Brian Hayden noted in his recent study of cultural notations, “record keeping devices have not held much interest for most ethnographers . . . Actual ethnographic illustrations or photographs of notational devices are exceedingly rare, and even more so [are] the explanations of what specific marks represent, why marks were organized in various ways, or the time period over which marks were made.”54 Thus, while we can say with confidence that forms that 54

Hayden, 2021, p. 2.

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     accumulate like tallies are generally interposed between the use of the fingers and manipulable forms that both accumulate and group, we cannot say with equal confidence that a collection of notches meant numbers, unless it contains the kind of overt numerical organization that does not emerge until the clay impressions of mid-fourth millennium Mesopotamia. By studying contemporary number systems, we gain an understanding of the emergence and elaborational processes that we can apply to the past, and this should guide how we interpret the archaeological record. We will start by looking at what material forms have to do with numbers in the next chapter.

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

Materiality in Numbers

In this chapter, we will look at how and why using material devices to represent and manipulate numbers acts as the mechanism of numerical elaboration. Essentially, material forms make quantity tangible, and tangibility lets us manipulate quantity into increasingly explicit forms and complex arrangements.1 Different types of material forms then have different properties for representing or manipulating numbers: Some are fixed and thus suitable for recording, while others are mobile and thus suitable for calculating. Properties may also act as limitations that can motivate the recruitment of a new material form, which is selected because it can do things that the earlier form does while addressing its limitations in some manner. An example is the tally, which accumulates in the way the fingers do, though its higher capacity also means that it can reach quantities whose visual indistinguishability can motivate the use of a form that can be rearranged into groups. Numerical elaboration thus becomes a matter of whether devices are used for numbers, which ones are used, and how they are used. We will also take a look at how material forms enable conceptual change through a model called conceptual blending. In this (neurocentric) model as originally proposed in the late 1990s, inputs from different mental domains interact and combine in a mental space to form new concepts.2 More recently, a material component was added to the model to act as an input domain and anchor the concepts it informs; this addition also shifted the locus of conceptualization from inside the brain to an enactive space found in the

1 2

Malafouris, 2010a; Coolidge & Overmann, 2012. Fauconnier, 1997; Fauconnier & Turner, 1998, 2002.



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     interaction between brain, body, and world.3 Once materially anchored, conceptual blending offers a mechanism whereby material properties act as proxies for numerical properties, thereby becoming conceptual ones. Material properties that become conceptual ones are affordances4 or exploitable attributes of the material forms used to represent and manipulate numbers – things like discreteness, linearity, capacity, manipulability, and conciseness. Finally, we will explore some generalized effects of material culture, which has more influence on numbers than motivating them through things to count and reasons to count them, or elaborating them through the properties of the devices used for representing and manipulating. Importantly, material culture also provides resources that enable and facilitate our creativity: an environment full of material forms that potentially can be recruited for numerical purposes5 or spark ideas and innovation.6 We will also look at some of the roles that material forms have in numbers, like accumulating and distributing cognitive effort across individuals and between generations, and how conditions of common use enable material forms to change and yet remain synchronized to average psychological, physiological, and behavioral capabilities.

         Numerical elaboration can be envisioned as a design space with three input domains: the brain, the human body, and the material world. These components all contribute different things. The brain contributes the abilities to appreciate quantity, “know” the fingers, and form concepts, and it neurologically integrates them in a way that makes it likely the fingers will be used for numerical expression.7 The body contributes fingers capable of being used both to manipulate objects

3 7

4 5 Hutchins, 2005. Gibson, 1977, 1979. Malafouris, 2021. Penner-Wilger et al., 2007; Piazza & Izard, 2009.

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6

Kirsh, 2014.

    and act as a counting device themselves. Material forms instantiate quantity and, when used to represent and manipulate numbers, provide properties that inform the resultant concepts. Across cultures and societies, the first two input domains – the brain and human body – are universal across the human species, and this accounts for the high degree of commonality observed across number systems, even those widely separated by distance and time. When we look at how number systems vary cross-culturally, it is easy to lose sight of the fact that numerical organization and structure are in fact highly circumscribed, when compared to the variability found in language and culture. No number systems, for example, are mutually unintelligible in the way that different languages can be. For its part, the material domain permits some latitude of choice in whether devices are used, which ones are used, and how any particular one is used. This latitude accounts for much, if not most, of the variability between cultural number systems and within any particular number system when considered over time. Cross-culturally, five basic types of material forms are used to represent and manipulate numbers (Table 7.1): distributed exemplars, the hand, devices that accumulate, devices that accumulate and group, and notations. As previously noted, distributed exemplars are features of the natural or cultural environment that typify a particular subitizable quantity, like the arms exemplify two for the Mundurukú. The hand is an embodied device for expressing subitizable quantities and scaffolding the first ones that are not subitizable, five and ten. Devices that accumulate include the tally, knotted strings, and torn leaves, while devices that accumulate and group include the counting tokens used in Mesopotamia and the knotted strings or khipu used by the Inka. Notations are written marks that represent quantity with the greatest conciseness of any form. Assignment to one of these categories is not always clear-cut. For example, depending on how it is used, the hand can be considered a distributed exemplar, or a device that accumulates, or a device that both accumulates and groups. Similarly, a tally is both a device that accumulates and a form of

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     Table 7.1 Types and chronology of material devices used in numbers. Starting with numerosity and material forms whose quantity is perceptible and manipulable, numbers become elaborated through the properties of the devices used to represent and manipulate them; new devices are selected on the basis of shared functions and the capacity to address the limitations of previous forms. Versions were previously published in Overmann (2018a, Table 1, p. 469; 2019a, Table 1, p. 10; and 2019b, Table 11.1, p. 212). Numerical device

Descriptions, solutions, and structure

Distributed exemplars  Two: eyes; arms; deer footprint  Three: tripod; bird claws; bird footprint; rubber seed; pronged fishing arrow  Four: spotted animal skin; brother (to three)

 Description: Commonly encountered natural and cultural objects whose quantity is appreciable, reliable, and expressible iconically (e.g., by means of a display of the fingers or a phrase describing an exemplar) or indexically (with a gesture toward an exemplar or a phrase drawing attention to it).  Limitation: Expressible quantities are typically limited to the subitizing range, and the methods of expressing them are ephemeral. Exemplars do not comprise a contiguous (single) device, so they do not significantly influence numbers toward organization or structure.  Solution: A material device (e.g., the fingers) that can make quantity percepts tangible and manipulable, transcend the subitizing range, and influence numbers toward organization and structure.  Persisting structure: Forms and features of numerical language that conform to the limits of quantity perception or have etymological roots in material objects with subitizable quantity.

The hand  Fingers (bent, flexed, tapped, etc.)

 Description: The hand is the first contiguous device used for numerical representation because of the neurological

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    Table 7.1 (cont.) Numerical device

Descriptions, solutions, and structure

 Finger segments or joints  Spaces between fingers

interconnection between the parts of the brain that appreciate quantity and “know” the fingers, and because the hands are readily available, visually salient, and easily used for expression. The fingers are used either directly to instantiate and display quantity or indirectly to gesture at an exemplar of quantity.  Limitation: As a material device, the fingers provide little persistence and have limited capacity.  Solution: Devices capable of doing what fingers do (e.g., accumulate with linearity and order) but which also address their lack of persistence and capacity.  Persisting structure: Discreteness, linearity, stable order, ten-ness.

Devices that accumulate  Notched tallies  Knotted strings  Torn leaves  Marks on surfaces  Pebbles or corn  The human body  Collaborative fingercounting

 Description: One-dimensional devices accumulate to amounts that exceed the fingers’ capacity. They also persist longer than the fingers, with duration governed by durability of the substance used: Bone persists longer than wood, fiber longer than leaves, etc.  Limitation: Quantities higher than about three or four are increasingly indistinguishable (a limit inherent in the perceptual system for quantity), necessitating that items be matched to known exemplars or recounted.  Solution: Devices capable of doing what one-dimensional devices do (e.g., accumulate with capacity and persistence) but which also address the

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     Table 7.1 (cont.) Numerical device

Descriptions, solutions, and structure problem of visual indistinguishability through grouping.  Persisting structure: Accumulation, capacity, persistence; public in a way that bodies tend not to be; harnesses the power of material objects to accumulate and distribute cognitive effort.

Devices that accumulate and group  Counting boards and calculi (jettons)  Mesopotamian tokens  Abacus  Inka khipu  Collaborative strategies  Sorting strategies

 Description: Two-dimensional devices accumulate like one-dimensional devices do, but they also bundle the accumulated elements, either as appreciable quantities (one to three or four) or as amounts conforming to a well-exemplified quantity (e.g., often five or ten, the number of fingers on the hands).  Limitation: Loose elements lack integrity of form and are indistinguishable in higher (non-subitizable) quantities. The first may prompt enclosure, which contains but removes access to the elements; the second may inspire replacement by conventional forms understood as bundled values, which reduces the number of elements but requires the user to learn the bundling conventions. Khipus lack manipulability, so they cannot be used as a technology for calculation.  Solution: Devices capable of what twodimensional devices do (e.g., group) but which add integrity of form. For manipulable forms, a fixed technology for recording is added (e.g., in Mesopotamia, bullae and then notations were used with tokens). For fixed forms, a manipulable form for calculation is added (e.g., in the

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    Table 7.1 (cont.) Numerical device

Descriptions, solutions, and structure Inka system, counting boards and yupana were used with khipus).  Persisting structure: Grouping (exponential structure), relations, manipulability.

Notations  Handwritten notations

 Description: Notations accumulate and group like two-dimensional devices but add integrity of form. They are also concise, which increases the density of simultaneously viewable elements and enables relational data to be recorded at volumes far exceeding those possible with earlier technologies. If they are handwritten with sufficient repetition, learned relational data and the neurological reorganizations associated with literacy will influence numbers toward being reconceptualized as relational entities.  Limitation: Notations are fixed, so calculating must be performed manually until it can be supplemented with algorithms based on mental knowledge and judgments to material recording (e.g., long division performed using paper and pencil).  Persisting structure: Conciseness, entitivity, conceptualization of numbers as a system of relational entities.

notation; it can also be a device that both accumulates and groups, if grouping is foreseen and imposed as the notches are made. The types and chronology of material devices used in the cognitive system for numbers were derived from two primary resources.

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     The first consisted of ethnographic and linguistic data that included societies, cultures, and languages from around the globe, as well as both contemporary and ancient number systems. The second resource was an analysis of the properties or affordances of the various material forms as described in the ethnographic data. As defined by the ecological psychologist James Gibson,8 affordances are capabilities that 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. The term as used here is narrower in its scope: Affordances are properties of material forms that make them suitable for representing or manipulating numbers, or which cause them to interact in specific ways with the other components of the system. Devices are either fixed (like the notches on a tally) or manipulable (like pebbles or abacus beads), and this in turn permits or inhibits their use for recording (fixed) or calculating (manipulable). The recruitment of new devices is then systematized by the common starting point and the predictable interplay of device capabilities and limitations, influencing similarity in the devices recruited crossculturally. These factors produce a material chronology that spans and connects the perceived quantity of distributed exemplars to the symbolic meaning of numerical notations. Material devices occasionally form unambiguous sequences. An example is found in Mesopotamia, where the archaeological record provides evidence that notations emerged from and temporally followed tokens. In other cases, the evidence did not correspond to the time period in which the device would have been used, and such temporal mismatches required plausible explanations. Mesopotamia is again a good example: Finger-counting, which emerges early in the chronology of any number system (Table 7.1), is attested by analyzable forms of Sumerian lexical numbers. But evidence of the spoken forms of numbers does not emerge until the early third millennium BCE, while finger-counting and associated lexical names would have 8

Gibson, 1977, 1979.

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    emerged during the Palaeolithic. There are reasons why the evidence for finger-counting lags so far behind the time period of its initial emergence and use: Numbers had to be recorded in such a way that their spoken forms were preserved. Notations do not do this, which is the difference between “7,” a written notation that does not express sound values, and “seven,” a written form that does. As a result, several technological developments had to occur first. Writing had to be invented, and then techniques for recording sound values had to develop, and then reasons to apply those techniques to lexical numbers had to emerge. These technological developments plausibly explain the temporal mismatch between the third-millennium lexical evidence of finger-counting and its Palaeolithic emergence in Mesopotamia numbers. Distributed exemplars were placed first in the chronology because they are found in emerging number systems that count no higher than three or four, and because such subitizable quantities are appreciable through the perceptual system for quantity. Fingers were placed next because they too are found in emerging number systems – those adding five and ten to the subitizable numbers and those counting to ten and twenty with the fingers and toes. The use of the fingers is also found in most number systems, either as an influence on basic numerical organization and structure (e.g., decimal; vigesimal) or as observed finger-counting, or both. Their influence on organization and structure suggests that fingers are used early in the chronology of devices, while their use in both emerging and elaborated number systems suggests that older devices are retained, rather than replaced, as newer devices are incorporated. Notations were placed last as the dominant form of representation used in the most highly elaborated number systems known today. The remaining technologies, devices that accumulate and devices that accumulate and group, were placed intermediate to the technologies associated with emerging and notationally mediated systems, as consistent with their association with elaborated systems of numbers without notations.

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     Within the chronology as shown in Table 7.1, the order of devices is also internally consistent, as assessed by five trends:  First, the interplay of capabilities and limitations explains why new devices are recruited and predicts the form that new devices will assume.  Second, as material forms are recruited into the cognitive system for numbers, numbers become more elaborated and contain more properties, and this is consistent with and empirically verifiable through observations in the ethnographic data.  Third, representations become increasingly concise. For example, it takes the hands of eight people to represent the number seventy-three, but only a single stick or string with 73 notches or knots. With grouping, the amount of representations needed drops to ten – seven items indicating groups of ten and three single items – as would be the case for the Mesopotamian tokens and the Inka khipu. With notations, the amount of representations falls below ten characters, depending on the notational system employed:

in

Mesopotamian cuneiform; LXXIII in Roman numerals; 73 in Western numerals.  Fourth, conciseness is achieved by minimizing the explicit component – what the representational form actually is; the information it specifically displays – which attenuates through an increase in an implicit component that consists of the knowledge users need in order to understand the representational form. For example, “7” is concise, but its numerical meaning must be learned, as it cannot be realized by counting individual elements; in comparison, “|||||||”is countable and does not require learning a convention, but it is not concise.  Fifth and finally, numerical concepts become distributed over multiple forms, which makes them increasingly independent of any particular form. For example, emerging number systems that count to twenty are closely tied to the fingers because their use involves the fingers and their names are lengthy phrases describing the fingers.9 These numbers are tied to the fingers in a way that the Western numbers one through twenty are not because they can also be represented by things like notations and tallies, in addition to the fingers.

The chronology provides insights, if not principles, that apply to the devices employed by a particular society and which can explain 9

Miller, 1999; Silva, 2012.

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    the resultant organization and structure of its number system, no matter how different its qualities might seem at first glance. We will explore some exceptional systems in upcoming chapters, where we will see that they support the conclusion that the material forms used in representing and manipulating numbers significantly influence numerical content, organization, and structure. The chronology shown in Table 7.1 is highly systematized by a number of factors, starting with the common starting point. Subsequent developments in the design space are governed by the interaction of the input domains, particularly in what devices can and cannot do. If the society needs numbers to do something that the current material form does not readily enable, then a new form may be recruited into the cognitive system for numbers. As noted earlier, the new form will be selected on the basis of the properties it shares with the previous form, as well as its ability to provide capacities that the previous form lacks. The fingers, for example, provide contiguity as a coherent device that distributed exemplars do not, and they typically permit counting up to ten, while distributed exemplars are limited to subitizable quantities. Fingers are not suitable for recording, nor indeed for display for any length of time, since the hands are eventually needed for other purposes. If the hands fully satisfy a society’s need for numbers, they will be used until there is a need for greater persistence or capacity than what they can provide. At some point, the fingers’ limitations of persistence or capacity may motivate the incorporation of a device like a tally that shares with the fingers the ability to accumulate, while providing the persistence and capacity that they lack. It should be noted that new forms seldom replace older ones; rather, older forms tend to be retained and used side-by-side with newer forms, though perhaps with less frequency. An example is finger-counting, which persists crossculturally, regardless of the degree of numerical elaboration. In emerging systems that count to twenty, the fingers are often the only device for representing numbers, so they are involved in most, if not all, occasions where numbers are used. In contrast, in the Western

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     cultural tradition, notations are the dominant material form for representation, and fingers are used to a much smaller, though not insignificant, extent. As new devices are incorporated, the properties explicitly represented by previous forms may take implicit form. Some of the implicit component is conceptual: beliefs about what numbers are; expectations for what they can do and how they should behave. Some of the implicit component is habitual.10 In the context of incorporating new devices, habits acquired in operating a previous device can influence the way a new one is used, independent of whether or not the physical substance actually requires it. For example, the numerical impressions used in Mesopotamia in the fourth millennium BCE were organized linearly, from left to right, in descending order of numerical value (Fig. 7.1). This suggests that the devices used to make the impressions, the tokens, were organized similarly, even though this organization is not inherent to loose objects like tokens. The idea that tokens were organized and ordered is plausible because these qualities would have made the numerical information they represented more accessible, compared to random placement. Nothing in the material form of tokens would have necessitated that they be magnitude ordered in linear fashion, since they were loose, manipulable, and could have been ordered in any number of ways. Since their physical substance did not require it, tokens were likely ordered on the basis of expectations and habits acquired with predecessor technologies like tallies and the fingers. Each step in the chronology represents a technological distance that is relatively small and thus fairly attainable. Technological distance is a continuum of similarity in form and function between the current device and one that potentially solves its limitations. At one end of the continuum is a replacement device whose form and function are identical to the current device in all respects; of course, the identical device will not solve the current one’s limitations. At the far 10

Duncker, 1945; Birch & Rabinowitz, 1951.

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   

. . Administrative tablet (Sb 22218) from Susa, modern Iran, provisionally dated to 3500–3350 BCE (cdli P008218). The numerical signs on the obverse face are arranged, from left to right, in descending order of magnitude (in the ŠE system Š for grain, one N45 equals ten N14, and one N14 equals six N01; see Nissen et al., 1993). This order implies that the tokens used to produce the impressions would have been organized similarly. Photograph by Jean-Vincent Scheil (1923) originally published by Ernest Leroux, Paris.

end of the continuum is a device that differs from the current one in every respect; the case is hypothetical, since some similarity of form and/or function is presumably needed in order to make a replacement suitable for the intended purpose and the innovation needed to attain it achievable. Between these extremes, some amount of difference in form and/or function will separate the current device and potential solutions. When the technological distance is small, potential solutions

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     will inhabit the narrow range of objects that are similar in form and function; this appears to be the case in numbers, when the devices used for representing and manipulating are considered crossculturally. As technological distance increases, potential solutions become increasingly dissimilar in form and function, and the difficulty of envisioning them as suitable choices will increase as well. For example, the fingers’ limitations of capacity and persistence are unlikely to be solved by notations, a device whose form and functions are too dissimilar from those of the fingers to make the required innovation easily achievable – particularly as they presuppose properties like grouping, manipulability, conciseness, and conventions that have not yet been realized through devices like tallies and tokens that involve smaller technological distances. The net effect of the chronology of material forms in Table 7.1 is a concept of number with an extensive set of properties acquired from the different material forms used for its representation and manipulation. Especially as expressed in the form of symbolic notations, this concept involves a significant technological distance from the initial perception of quantity. This distance represents an amount of cumulative innovation that would be difficult, if not impossible, for any one individual or generation of individuals to achieve, except through the incremental steps realized by the systematized recruitment of intermediate material forms. The chronology of material forms also influences the emergence of arithmetical operations. Addition tends to emerge first, likely because all the material forms listed in Table 7.1, particularly those early in the chronology, permit accumulation. Subtraction, in comparison, does not seem to emerge until later, and this too is likely because few of the material forms used early in the chronology allow things to be separated enough to form ideas about subtraction; distributed exemplars, the fingers, and tally notches are good examples of the difficulty of feasible removal, though for different reasons. Of course, which operations emerge, and in what order, are influenced by the specific material forms used, how those forms are used, and

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    what a society needs its numbers to do in managing its sociomaterial complexity.

   All material forms used for numbers fall on one side or the other of a fundamental distinction: They are either good at representing numerical information, or they are good at manipulating it, but they are not good at both. This is a primary reason why new material forms are recruited into the cognitive system for numbers: the need to represent, or to manipulate, when the form(s) being used does not do this. Representing numerical information means displaying it. The fingers and an abacus can both be used to represent numbers, but as long as that representation takes place, the device cannot be used for anything else. When someone uses his fingers to represent the number four, he cannot also use his hand to pick something up, or write, or whatever. Some of his resources of memory and attention are also required, since he must remember to keep his hand in the position that displays four and then monitor himself to make sure that he does not forget and let his hand relax. Similarly, as long as an abacus displays a number, it cannot be used for calculating, since calculating requires moving the beads, and moving even a single bead changes whatever number is being displayed. An abacus is really good at manipulating numerical information by moving it around in ways that rearrange it. A tally is not good at manipulation, but it has some limited potential in that it is possible to group the marks as they are accumulated, or to orient them differently, or to make them different sizes. Even so, once a tally mark is made, it stays. In Western numbers, we get around the fact that written notations are fixed by writing in the interim steps and sometimes crossing them out and rewriting them, a form of moving them as physical elements. Calculating with notations also requires making mental judgments that involve the knowledge of numerical relations, as for example, knowing that 3 times 6 is 18. It also involves the knowledge

of

algorithms

and

conventions

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for

rewriting.

In

     multiplication, for example, 123  456 starts with multiplying the units (3  6), an interim step that is reflected by writing 8 in the units column and carrying 1 into the tens column. In subtracting 9 from 25, the 2 in the tens column is crossed out and a 1 written in, and another 1 is written in the units column to make 15, an amount from which 9 can be subtracted. So notations are effectively manipulable, but their actual forms are fixed once we write them and must be changed by writing interim steps, crossing out, and rewriting. Since it is difficult to find a device that does both, what typically happens is that a device that is good at representing is used alongside another that is good at manipulating. In the Middle Ages in Europe, counting boards with counters (calculi or jettons) were used for calculating, while Roman numerals were used to record the results.11 In what is now modern Peru, the Inka used complex configurations of knotted strings known as khipu to record their numbers, but an abacus-like device called a yupana for calculating (Fig. 7.2).12 While modern calculators and computers combine these functions, they have not managed to escape the problem of being unable to do both simultaneously. They are very good at both calculating (manipulating) and displaying (representing); however, representing must be offloaded in some fashion, or the device is tied up by displaying the result, and while calculating is quite fast, the device cannot display a number as long as manipulation is in progress.

    How might the properties of the devices used in representing and manipulating become conceptual ones? In the late 1990s, the cognitive scientists Gilles Fauconnier and Mark Turner proposed a model of conceptual blending in which mentally represented inputs interacted in ways that produced novel outcomes. Essentially, distinct mental structures – concepts, thoughts, understandings, abstractions – 11 12

Pullan, 1968; Stone, 1972; Evans, 1977; Reynolds, 1993. Ascher & Ascher, 1981; Brokaw, 2010; Urton, 2010.

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   

. . Chief accountant holding a khipu, a knotted-stringed device for representing numbers used by the Inka. The yupana, or Inka counting board, appears to the bottom left. Drawing 143 (p. 360) of Guamán Poma’s El Primer Nueva Corónica y Buen Gobierno [The First New Chronicle and Good Government], written around 1615. Image in the public domain.

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     were combined mentally in a way that could generate novel mental structure.13 The model had four interacting 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. The blend was envisioned to inherit some of its structure from the input spaces, with the interaction of the various inputs having the potential to produce novel structure as well.14 An example is the blend of numbers with the concept of two-dimensional space; blending was the mechanism whereby numbers acquired spatial attributes like angles, magnitudes, and coordinates.15 A couple of caveats need to be highlighted. First, neuroscience does not really understand what a concept is. In fact, how physical phenomena like electrical impulses, neurotransmitters, and synaptic responses yield qualitative aspects of experience, semantic meaning, and consciousness remains one of the central mysteries in studying the brain. In this regard, the conceptual blend is more functionally descriptive than actually reflective of how conceptualization works. A more fundamental problem is that all of the model’s inputs, outputs, and processes are mental. When the brain is seen as solely responsible for conceptualization, cross-cultural variability in number systems becomes a matter of some societies being better equipped than others which is surely not the case.16 The original model is consistent with how we tend to conceive the brain and world, which is in dualistic terms. Putting the brain in one category and the world in another is consistent with our experience of material objects, and it is undoubtedly beneficial for controlling variables and confounds during neuroscientific experiments. Nonetheless, separating mind and matter is not the only way the boundaries of cognition can be drawn. Nor is it necessarily the best 13 14 15 16

Fauconnier, 1997; Fauconnier & Turner, 1998, 2002. Fauconnier, 1997, pp. 149–151; Fauconnier & Turner, 1998. Fauconnier & Turner, 2002, p. 243. Lévy-Bruhl, 1910; Piaget, 1928; Chomsky, 1980, 1982; Hurford, 1987.

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    perspective, if our goal is understanding cognitive change across cultural and evolutionary scales of time, where the material domain is both witness and causal agent. Minimally, our model of conceptualization needs to include a mechanism whereby our physical experience of the material world informs the concepts we produce about it. For surely our experiences of physical properties like weight and quantity inform our concepts of them, as well as the material systems we create to represent them. Moreover, because they are aspects of the world regardless of whether we conceive or materially represent them, physical properties necessarily precede our concepts and material devices.17 They also continue to inform them, since properties like weight and quantity do not vanish or change simply because we have formed a concept of them or represent them with a material form, nor do our concepts and material representations change in ways that suggest they have become unmoored from the properties they instantiate. Thinking about the relation between material heaviness and conceptual units of weight, the archaeologist Colin Renfrew noted “an inherent link between the physical and material and the symbolic. . . . Units of weight [and number] are indeed conceptual, but they would be unthinkable without the experience of the physical reality of weight” and quantity.18 Thus, conceptual blending needs a material input, one that can account for the priority and continuity of the material domain, as well as its influence on the concepts we form about it. Further, the model should also include the material domain’s capacity for manipulation and parsing, as this enables us to explicate and elaborate our concepts. And ideally, the model should achieve all of this in a nondualistic manner. Apparently thinking along these lines, the cognitive anthropologist Edwin Hutchins changed the model for conceptual blending by combining the original four mental spaces into a single mental domain and adding a material domain as an input (Fig. 7.3).19 These 17

Renfrew, 2004, p. 23.

18

Renfrew, 2001, p. 98.

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19

Hutchins, 2005.

    

. . Materially anchored conceptual blending. Mental and material inputs combine within an enactive space that is not brain-bound. The input spaces are densely interconnected by mapping and projective mechanisms that are prioritized and guided by attention (see Hutchins, 2005, p. 1561; Malafouris, 2013, p. 99). Projection establishes connections of “identity, analogy, similarity, causality, change, time, intentionality, space, role, and part-whole, and in some cases . . . representation” between mental and material domains (see Malafouris, 2013, p. 100). Projection is triggered and supported by external material forms that interact with, and influence, psychological and physiological capacities, knowledge and expectations, and behaviors (see Malafouris, 2013, p. 102). As they did in the original model, conceptual blends inherit some of their structure from these inputs, and novel structure can emerge from the interaction of the inputs. Adapted from Malafouris (2013, Fig. 5.2, p. 101). Previously published in Overmann (2019b, Fig. 3.1, p. 39).

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    mental and material domains project to an enactive space where blending occurs. This locus is found in the interactivity between brain, body, and world (the idea of enactivity will be developed further in the next chapter). Just as they did in the original model, blends inherit some of their structure from the inputs and can generate emergent structure. But material forms act to anchor and stabilize concepts, and their properties have the potential to inform conceptual structure. In essence, concepts not only involve material structures but are partly constituted by them. Material properties can inform conceptual properties, and material manipulations can provide opportunities for conceptual change by giving brains the chance to do two things they are really good at: recognizing patterns and forgetting details. In the modified model, material forms acquire roles in initiating, persisting, and modifying human concepts. Once materially anchored, conceptual blending gives numbers a set of properties originating in the different devices used for representing and manipulating. As we will see in upcoming chapters, numbers acquire linearity, discreteness, stable order, and grouping by fives and tens from the fingers; they acquire capacity and persistence from tallies; they acquire manipulability from forms like pebbles, tokens, and counting boards; and they acquire conciseness, relationality, and entitivity from notations. Through the implicit component of numbers, these properties persist across the inclusion of new devices, something that ultimately makes them functionally independent, not just of any particular device but also of the total set of devices used to represent and manipulate them. The implications go beyond even this, for numerical emergence and elaboration show materiality as having an active role in shaping thinking and the mind itself. This is discussed further in conjunction with material agency in the next chapter.

     Considered in general, material culture provides us with objects that can be recruited for solving problems. Faced with a new problem, we

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     may have some idea of what we need to do to solve it because it is something that we want to accomplish that our current device does not provide or enable. In general, if our current device does not do what we need it to do, solving the problem means either modifying it or choosing a new device that can satisfy the existing functions while adding new ones. In this case, a different material form may strike us as suitable for the purpose we have in mind, or be easily adapted to it, representing a small technological distance. An example is found in the hand and the tally. As we cannot add more fingers to our own hand, the need for greater capacity than what the fingers can provide may motivate us to recruit the hands of other people; this adds capacity, but not persistence, leaving a limitation that at some point may motivate another material solution with greater persistence, like the tally. Alternatively, we might choose a device like the tally that can accumulate like the fingers do; while the technological distance involved is greater than adding the hands of other people, the solution adds both capacity and persistence. Which of these two strategies might be more typical is unknown. Cross-linguistic terms like one finger on the second man for twenty-one in vigesimal systems20 suggest that cooperative finger-counting is common, while terms meaning two tens and one do not exclude the possibility that cooperative counting was used prior to adopting a device like the tally. Shared form and/or function includes an implicit component that consists of the knowledge, expectations, beliefs, habits, and skills required to use and understand the current material form(s) used for numbers. The implicit component predisposes us to use a new material form for numbers in the same way we use the old form. For example, fingers and tallies accumulate in a sequential and linear manner that can influence how later forms like tokens or pebbles are arranged, even though loose forms can be arranged in nonsequential, nonlinear patterns (circles or triangles, for example). Importantly, the predisposition to use new forms in the same way as older forms 20

Nansen, 1893.

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    acts to transfer conceptual structure between material forms as new ones are recruited into the cognitive system for numbers, providing a conceptual continuity vital to the process of elaboration. This predisposition also acts to narrow the range of potential solutions, which may increase the technological distance needed for innovation. These factors, in turn, limit and thus systematize the forms that tend to be recruited cross-culturally. While the range of potential solutions is narrowed to those material forms whose forms and/or functions bear significant affinity to current solutions, material culture in general can also function as a distraction, shaking loose the effects of familiarity and functional fixedness, the tendency to use forms the way we learned to use them. Habit, custom, expectation, and knowledge – the implicit component of numbers – not only narrow the range of possible solutions, but also blind us to possibilities for change. Familiarity is “the enemy” of creativity and innovation 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.”21 But material culture, writ large and expanding under conditions of socio-material increase, provides a landscape of possible forms and functions. These can function as random stimuli with the potential to distract us,22 thereby shaking loose the familiarity, habits, and functional fixedness that narrow the range of possible solutions, and setting our creative and pattern-completing mechanisms free to find solutions that might involve greater technological distances. Here at last we might also find an answer to the question of what it is about material culture in general that stimulates numerical creativity, the raison d’être for the correlation between socio-material complexity and numerical elaboration that has not been found in subsistence strategies or specific resources.23 It is not just that sociomaterial complexity motivates us to manage it, or that materiality 21

Kirsh, 2014, p. 22.

22

Kirsh, 2014.

23

E.g., Divale, 1999; Epps et al., 2012.

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     and numbers become habitual and preferential problem-solving resources and strategies, or even that devices enable us count to the higher numbers that potentialize numerical relations, though all of these are true. Rather, it is that material culture provides us with resources and options that enable us to innovate beyond the narrow range of the familiar, by distracting us from it and stimulating our creativity.

     Numbers in the Western tradition have become distributed over multiple devices: notations, the form we use the most often; fingers, whose informal use in developmental acquisition persists as adult use in counting; tallies, something we might resort to on those occasions when we need to count more things than we can easily accomplish with our fingers, or we need a more permanent record of that count than what our fingers would permit; and tokens, manipulable things like coins that evoke the thousands of years in which counting boards were used. This distribution makes our conceptual numbers functionally independent of any and all of these material realizers, something that is not true, for example, of emerging numbers, whose use and expression are inextricably tied to forms like the fingers. Distribution and independence make the material component largely invisible to us, so if we think of the material forms used for numbers at all, we think of them as passive repositories for mentally created numbers. As for those mental creations, they are generally thought of as having emerged fully formed, with all their many properties, as Athena was once said to have sprung from the forehead of Zeus. The view of numbers as wholly mental creations is shared, whether numbers are seen as biological or linguistic capacities, as concepts introspectively intuited or as formed from embodied experience, or as eternal, immaterial entities that we somehow come into contact with. However commonly held, this perspective ignores the slow and incremental history – and prehistory – of using material forms to realize and elaborate concepts of numbers, all the way back

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    to the point in the Palaeolithic where numbers began as appreciable quantities expressed with fingers and gestures. Even if we do not have all of the details for exactly when and where this occurred, the material domain is nevertheless crucial, for it accumulates and distributes cognitive effort between generations.24 The task of each new generation becomes simple: It does not have to invent numbers ex nihilo; rather, it merely needs to learn and use the tool(s) for numbers in whatever form(s) it currently has. Learning the tool changes behaviors and brains, opening up possibilities for subsequent change in material forms and functions as tools are used. Realization and elaboration, while slow and incremental, become achievable simply because they involve the sustained, collaborative use of material forms. In this way, materiality brings the realization of counting sequences, simple arithmetic, and complex mathematics within the grasp of societies composed of average individuals. This is not to say that the process excludes individual innovation, for someone, somewhere, was surely the first person to notch a stick or stripe mud on an arm in a way that meant quantity. Rather, it is to say that individual innovations are part of a larger evolutionary process that ultimately yields highly elaborated forms: material devices like symbolic notations whose form and function are so far removed from the perceptual experience of quantity that they cannot possibly be invented from scratch, no matter how inventive an individual might happen to be. This is where the material form finds its power – its agency – for it accumulates the cognitive effort of past individuals and generations and distributes it to new individuals and future generations. Essentially, numbers are not the outcome of a single brilliant act of creative genius, but rather, the mundane result of the collaborative use of material devices sustained over thousands of years or more.

24

Hutchins, 2005.

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    

      Material devices change under use. They change when they are used because they are used, and their change is a consequence of how they are used. Here we want to think about what happens when a device is used in common by multiple people – not different people passing the identical device from hand to hand, but different people using essentially identical versions of the same device, like everyone in a group making tallies or writing numerical notations. Using a material form can influence change in the behaviors and psychological processing, and using a material form in common can influence the same changes in those members of the social group who interact with it. Common change among the group members can opportunize further change to the physical device, however many versions of it are spread among the group. Common use can also cause certain features of the device to become emphasized: for example, those more effective in achieving a specific result, those requiring less physical effort, or those activating aspects of perceptual systems more strongly. Such changes do not necessarily happen all at once, and they are not associated with individuals isolated from the group. When devices are used in common and for generations, such changes can have dramatic outcomes. A good example of use influencing material form and intensifying usable features is literacy. Most people who have the opportunity to interact with the material form that is writing can master the skill called “reading,” which involves acquiring specific changes in behaviors and neurological processing, as for example, mastering handwriting and training the Visual Word Form Area of the brain, a part of the fusiform gyrus in the temporal lobe, to recognize written characters.25 Early writing for numbers consisted of elements repeated to express quantity and conventions that meant grouped quantities, while nonnumerical language was approximated through small pictures and conventions. All these eventually became script, abbreviated forms 25

Cohen & Dehaene, 2004; Dehaene & Cohen, 2011; James & Engelhardt, 2012; Nakamura et al., 2012; Perfetti & Tan, 2013; Vogel et al., 2014; Popenici, 2022.

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    that the visual system could easily distinguish as contrastive, and which could only be read by acquiring the specific neurological and behavioral changes associated with literacy.26 In historical writing, the process involved generations of incremental change in behaviors and brains, changes that the material form of writing accumulated and distributed to new users when they learned to interact with it. In contrast to writing, where a single form (including numerical notations) changes under common use, numbers change by recruiting new material forms and incorporating their properties as conceptual content and structure. In numbers, common use reinforces function and form in the devices used, and these in turn influence how numbers are acquired and understood. Another effect that use in common has is that material forms remain synchronized to common user abilities, even as they change. Again, literacy is a good example, as learning to read and write require no unusual attributes in abilities like fine motor control, eyesight, language, attention, or working memory. This is because common use would have had a leveling or averaging effect. That is, as the material form changed, it was subjected to aggregate psychological, physiological, and behavioral capacities that spanned the highs and lows of the population of users. Because highs and lows tend to cancel each other out, the net result was that the material form was able to change quite dramatically, yet remain synchronized to the capacities and abilities of the average user and thus remain usable in common. In the next chapter, we will examine the theoretical framework for these ideas, Material Engagement Theory.

26

Overmann, 2016a, 2022.

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

Materiality in Cognition

In this chapter, we will discuss the theoretical framework – Material Engagement Theory (MET) – used in analyzing material forms as a component of numerical cognition.1 MET is an approach to the study of material culture that assumes it plays a role in human cognition. MET is particularly interested in the roles that tools play in cognition, and how those roles would have influenced human cognitive evolution. In taking this perspective, MET differs from traditional archaeological and cognitive approaches to the study of the mind, both of which have tended to see the mind as something distinct and qualitatively different from the material world. MET has three central tenets. The first is the idea the cognition is extended and enacted. Extension positions material forms as a constitutive component of cognition, while enaction sees cognition as being the interaction between brain, body, and world. These positions radically reconceive the mind as more than neural activity inside the brain, and material forms as an active, constitutive component of the mind. This view transforms the archaeological record from a passive witness to the growing powers of the hominin brain to being an integral part of the ancient mind, thereby opening up a dimension of evolutionary change that is largely inaccessible to the methods and theories of the cognitive sciences. MET’s second tenet is material agency, the idea that material forms can influence our behaviors and brains. As an example, we will look at the ways in which material forms like writing accumulate and distribute incremental change in behaviors and brains, a notion distinct from the idea that writing accumulates and distributes knowledge. The third tenet 1

Malafouris & Renfrew, 2010; Malafouris, 2013.



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    is enactive signification, the idea that meaning emerges when we interact with material forms. Here we will distinguish the material sign, which is visible, tangible, and physically manipulable into new forms, from the linguistic sign, an ephemeral form with none of these qualities. We will see that redrawing the boundaries of cognition to include material forms has the potential to generate novel insights into the nature of cognition and thinking, and that these insights can be useful independent of whether or not MET’s three tenets are accepted. At the same time, we do not want merely to gain new insights into what material forms have to do with numbers. Rather, we want as complete and accurate an understanding of human cognition and thinking as possible, and examining the roles that material forms have in cognition and thinking can help to achieve this. In examining numerical cognition, this means setting aside the traditional model of numeracy as neural activity inside the brain in favor of a model of numeracy that includes the brain and the material forms and behaviors attested archaeologically, ethnographically, and linguistically. This does not mean that the traditional model is necessarily wrong; rather, it means that the potential conclusions of each model will differ according to what it includes or excludes. While we want to understand the mind as more than the neural activity inside the brain, this in and of itself is a recent historical development, as the brain has not always been seen as having any role in cognition or thinking. In ancient Greece, for example, Aristotle thought the brain was a minor organ that acted as a cooling unit for the heart.2 In ancient Egypt, the brain was so little regarded that mummification involved drawing it out through the nostrils with an iron hook, and whatever the hook could not reach was dissolved with drugs.3 The brain’s treatment contrasts sharply with the way the other organs were handled: The stomach, intestines, lungs, and liver

2 3

Aristotle, 1961; Gross, 1995. Godley, 1922; Wade et al., 2011; Wade & Nelson, 2013a, 2013b.

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     were thought to be needed in the afterlife, so they were removed and stored in jars. The heart was left in the body so it could be weighed by Anubis, the jackal-headed god of the underworld, as a measure of the value of its owner’s deeds in life. About two thousand years later, the French philosopher René Descartes formally divided mind and matter into ontologically different substances,4 a distinction known today as Cartesian dualism. In dualism, the mind is immaterial, and the body and world are material. This ontological division posed an immediate problem, which was how an immaterial mind might interact with a material body. Consistent with the era’s religious beliefs about an immortal soul, Descartes thought that the body received the subtle vibrations of the mind through the pineal gland, a small structure located in the center of the brain. As translated by the pineal gland, the subtle vibrations of the mind moved the body just like pulleys and strings move a puppet. By the early nineteenth century, the idea that the brain might have something to do with cognition and thinking began to gain traction. The German physician Franz Joseph Gall argued that the brain’s condition as a material substance was the very quality that enabled it to produce and exercise the mental faculties and traits.5 Gall’s valuable work in associating cognition and personality with the brain is often overshadowed by the pseudoscience it generated: phrenology, the now-outdated notion that bumps on the skull could determine and indicate mental abilities and character. If phrenology now seems fantastic or silly, identifying mind with brain was heresy in Gall’s time – a mere two hundred years ago. Treating the mind as part of the physical body denied its immateriality, and this essentially denied the existence of the soul, something that was intolerable to religious and secular authorities alike. As a result, Gall’s work was banned by the Church in 1802, and he lost his post and was expelled from Austria in 1805. 4

Descartes, 1637, p. 33; 1644, Part 1, Article VII; 1993.

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5

Gall, 1835.

    Within the past hundred years, it has become increasingly uncontroversial to associate mind with brain, so much so that today the two are commonly equated, if not synonymous. Here we want to separate them again, but not in ontological terms. Instead, we want to know what else the mind might include, in addition to the brain.

     MET views cognition as embodied, embedded, extended, and enactive (4E). Embodiment recognizes the body as having a role in cognition,6 while embeddedness and situatedness see cognition as influenced by aspects of the environment.7 As these views are relatively uncontentious, they will not warrant further discussion here. Extension and enaction are more radical positions. Extension holds that material forms are a component of cognition, along with the brain and the body, while enaction holds that cognition is the interaction between brain, body, and world.8 Extension has occasionally been mischaracterized and dismissed as panpsychism,9 the view that all things have a mind or a mind-like quality, or hylozoism, the idea that matter is in some sense alive. Here we will not only assume that panpsychism and hylozoism are incorrect in what they assert, we will also note that what they assert is distinct from extension. Rather than arguing that material forms themselves have minds or life, extension claims that material forms are a critical component of human thinking. Part of the difficulty in understanding extension is that we think of brain and world in dualistic terms – as ontologically distinct categories. A dualistic conceptualization is not just historically entrenched, it may also seem consistent with our own experience of mental content and material objects. Certainly, mental content seems

6

7 8

9

Lakoff & Johnson, 1999; Chemero, 2009; Prinz, 2009; Malafouris, 2017; Varela et al., 2017; Shapiro & Spaulding, 2021. Smith, 1999; Haselager et al., 2008; Overmann & Malafouris, 2018. Clark & Chalmers, 1998; Clark, 2008; Di Paolo et al., 2010; Gallagher, 2013; Hutto, 2013; Hutto & Myin, 2013. E.g., Johnson & Everett, 2021.

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     to connect and disconnect easily from any of the material objects that we happen to encounter. We can also recall and think about material objects without them being physical present, and their absence might suggest that material objects are not part of or implicit to our thinking. We could argue that thinking about absent material forms necessarily requires at least some experience of them, but there is an easier way to illustrate extension and enaction: reading and writing as cognitive states. As cognitive states, reading and writing do not – indeed, cannot – exist without the material component that is writing, as well as its active engagement. We may seldom think of writing as a material form, distinct from whatever it is written on, the instruments used to make the marks, and the interactions that recognize and produce them. This is partly due to writing’s flexible combinability; a set of characters concatenated into seemingly endless permutations does not have the same sort of solidity or permanency as, say, an iron wrench does. We have also been taught to think of the characters that comprise writing as symbols, prioritizing their ability to be meaningful over the material forms that they actually are. Writing as a material form has a historical chronology that lets us understand how an elaborated version like the one used to write these words emerged from the practice of handwriting small pictures some 5000 or 6000 years ago.10 Here we should note a slight confusion of language: Writing is both noun and verb, a conflation true of more languages than English that shows how closely the material form is associated with the productive act. For the purposes of this discussion, the term reading, the active visual engagement of written characters in order to grasp their meaning, will be assumed to encompass its productive counterpart, writing (the verb), the activity of using tools like pen and paper to produce characters. The term writing (the noun) will designate the material component, the result of the productive act. 10

Overmann, 2016a, 2021g, 2021a, 2023a.

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    Reading (and writing as verb) is an example of extended cognition because someone cannot read unless and until she interacts in a specific way with a specific material form, which is writing (as noun). She can recall what she has read, discuss it with someone else, carry the book from place to place, look at the cover of the book as it lies on her desk, look at a paragraph without engaging her attention or focusing and moving her eyes over the characters, or remember or imagine herself to be reading, but these engagements, recollections, and imaginings are not the same cognitive state as reading itself. To read, the person must use her hands and arms to hold and manipulate the pages, focus and move her eyes over the characters, and attend to the characters in such a way that they form intelligible words and meaningful sentences. The mental processing that occurs in the brain as she reads is undoubtedly important to the cognitive state that results, but so too are the material form of writing and the behaviors that actively interface the processes of the brain with the material form. Once we recognize reading as a cognitive state that necessarily includes a material component and which consists of the interactivity between brain, body, and material form, we have a plausible example of cognition that is extended and enacted. Further, once we accept reading as an extended/enacted cognitive state, we can ask whether it is the brain or the mind that assumes the cognitive state. Since the brain cannot assume the cognitive state unilaterally, we must consider the possibility that the mind consists of more than just the brain. If this is so, then the mind can be considered a system in which the brain is a component, along with material forms and behaviors. Accepting reading as an example of extended/enacted cognition admits the possibility of other occasions where the mind includes material and behavioral components in addition to the brain. Reading also reveals that we fluidly connect and disconnect from the material objects that are part of the mind, and that these are part of the cognitive system just so long as we engage them in certain ways. These circumstances in turn suggest that we likely engage material forms in a variety of ways with the potential to comprise a

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     cognition that is extended and enactive, with reading being a particular case that likely differs from other cases in terms of the specific psychological processes, behavioral movements, and material forms involved and the respective contributions of these different components. Reading as extended/enactive cognition also opens up a dimension of cognitive change over time in which material forms accumulate and distribute the incremental change in behaviors and brains realized through sustained, collaborative use of a particular tool. The material dimension of our cognitive change has particular interest in investigating the cognitive evolution of our species, as it is both reflective of cognitive change over cultural and evolutionary spans of time and causally implicated in it. Reading as an extended and enactive cognitive state has a specific implication for numbers: Because written numerical notations can participate in the extended/enactive state that is reading, and because numerical notations share function and form with precursor technologies like fingers and tally notches, then the cognitive states involving precursor technologies are also likely extended and enacted. As we have already argued that reading is extended and enactive, let us consider how written numerical notations share function and form with their precursors. Material forms represent quantity by instantiating it. Three fingers, three notches on a tally, three beads on an abacus, and three vertical strokes in the cuneiform and Roman numerals for three (Fig. 13.1) all mean three because they are three (conventional or ciphered forms like “3” emerge from instantiated forms like III; this is noted briefly below in the context of material agency and explored in Chapter 13). Fingers, tally notches, abacus beads, and numerals like

and III instantiate quantity because we

appreciate quantity through a perceptual system that is highly visual in nature. The critical point is that all these material forms of representing number – fingers, tally notches, abacus beads, numeral elements – have the same function and form: They all consist of three elements, and they all represent quantity by instantiating it. The implication

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    raised is this: Given that notational forms are implicit to the extended/enactive cognitive state that is reading and that numerical notations are part of the set of notational forms, the function and form shared by numerical notations and precursor technologies like fingers and tallies imply that precursor technologies are part of extended/ enactive cognitive states as well. As notations, fingers, tallies, and beads differ in their properties – notations and tally marks are fixed, abacus beads are manipulable, and all of them persist longer than the fingers, whose numerical representations are ephemeral – the cognitive states involving precursor technologies are likely to differ from the one involving notations in terms of the psychological processes, behavioral movements, and material affordances involved and the respective contributions of the different components. The claim that cognition is enactive – that it consists of the interactivity between brain, body, and world – needs some further discussion because it is essential to the idea that conceptualization is materially anchored and occurs in an enactive space. To illustrate enactivity and materially anchored conceptualization, consider the blind man’s stick.11 Someone who is visually impaired can navigate his physical environment by using a cane. The cane extends his tactile perception to its tip. When the tip comes into contact with the road, the man feels the surface of the road and its tactile characteristics through the cane. In this case, perception can no longer be said to be strictly inside the brain alone or the brain as embodied, since the stick is integral to the percept; neither can perception be said to be strictly inside the brain/body and the stick, since the tactile characteristics of the road lie outside them all. Perception, and indeed, the blind man’s conceptualization of his immediate physical environment, are effectively distributed between multiple spatial locations – brain, body, stick, and road – in a way that they exist in their interaction and in none of them individually. Perception and conceptualization are located in the interaction of the components of the cognitive system, 11

Malafouris, 2008; Merleau-Ponty, 2012.

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     and the cognitive state not only consists of the interaction, it also depends on all of the components being involved. This was true of the previous example as well, since reading consists of the interaction of psychological processes, bodily movements, and material writing; cognitive activity is distributed between the components in a way that it cannot be localized to any individual component, and it depends on all of them being actively involved in the system. The nature of enactivity is perhaps more intuitively apprehensible when it is couched in tactile terms because the blind man’s stick involves physical connections between solid components in a way that vision and reading do not. But it would be a mistake to conclude, from the plausibility of the tactile example, that enactivity necessarily requires physical contact between solid components. Alternatively, vision and reading can be conceived as involving physical connectedness between physical components: Rather than hands, fiberglass, and asphalt, air as a physical medium connects the cones and rods of the eyes with the light waves refracted by the surface properties of written objects. Another important aspect of enactivity is that perception ceases when motion does. The person navigating by cane feels nothing if he does not move the stick to explore the features of the road; the person with a book does not read it if she does not move her eyes in specific ways across the material form of writing on the pages. Perception depends on our active engagement of the world; action is the essence of perception.12 Concepts, in contrast, once formed can persist beyond the active engagements of the material forms that inform them, through mechanisms like memory and habit, and of course, the physical presence of the material forms that inspired, informed, and now anchor them. Redrawing the boundaries of cognition to include material forms and the interactivity between the components has several outcomes. First, as previously noted, it lets us gain new insights into the 12

Gibson, 1979, 1986; Noë, 2004.

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    nature of cognition and thinking, and we want as complete an understanding of these things as possible. Second, it acknowledges that our interactions with material forms change us behaviorally and psychologically – even morphologically – just as we change their function and form, even if our changes occur on faster timescales than theirs do. Learning to use a tool means acquiring knowledge and physical skills, things that involve synaptic changes in the brain and improvements in performing and controlling motor movements. Changes in knowledge and physical skill related to tool use, in turn, have the potential to change the tool’s function and form, and interacting with the resultant forms and functions is how other individuals acquire the same changes in behaviors and brains. Change in tool function and form in response to user behavioral and psychological change, while incremental and slow, still enables the material form to realize, accumulate, and distribute cognitive effort. A Palaeolithic genius did not conceive numbers one day and invent numerals the next; rather, generations of average individuals perceived subitizable quantity and expressed it in ways that involved material forms, which responded by transmitting the behavioral and psychological changes involved in using them to new individuals and generations. Material forms transform us, even as we transform them, while meaning emerges from our interaction.

   Agency is the capacity of an agent to act in the world. It is generally conceived as something humans have that is related to their abilities to form and pursue courses of action. Agency is sometimes called free will, and it often means intentionality. Consistent with our earlier rejection of panpsychism and hylozoism, we will assume that material forms lack free will and the kind of intentionality that we believe ourselves to have. Agency can also be understood as a relational property, something that causes change within a system, similar to the way that an affordance is a relation between agent and object capabilities. Material artifacts display agency in changing human

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     behavior: A speed bump, in offering the risk of damage and the attendant need for expensive repairs, insists that drivers slow down and thus co-creates their actions; a stone, in how it fractures when struck, influences and thus 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. The example of the speed bump can become entangled in discussions of where intentionality lies – in the roadway protuberance, in the person who installed it, or in the drivers who slow down their cars. Accordingly, we will briefly shift our focus from manufactured objects to natural ones in order to stipulate that material agency is not a 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 certain stone is found. Though neither equivalent to, nor interchangeable with, human agency, material agency is complementary and 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 are a function of its affordances. Affordances are inherently related to an agent’s abilities, both physical and mental; if a species lacks the ability to exploit a particular material property, that property is not an affordance for that species. A monkey would likely have little use for a calculator beyond mashing its buttons to display the lights or using it as a projectile. An empirical investigation of what monkeys might do with a similar but heavier device, a computer, found that they bashed it with a stone and defecated and urinated on it;14 after a month’s time, they had also produced a five-page text consisting of just over 13,000 characters, most of which (76 percent) were the letter “S.”15 Clearly, such employments also fall within the human behavioral repertoire, as do 13

Malafouris, 2010b, 2013.

14

AP, 2003; BBC, 2003.

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15

Elmo et al., 2002.

    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 a particular task without the affordances the calculator provides. Calculators and computers also show that affordances are not just features of the natural environment: They can be encoded as material artifacts, like the calculator encodes the ability to perform arithmetical algorithms. When affordances are encoded as artifacts, their capabilities influence behaviors. For example, 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.”16 Such artifacts also make it likely that current tasks and problems will be approached in certain ways. Navigation will likely be performed by means of the sextant, mathematical tasks by means of a calculator or similar instrument, rather than using or inventing some other method that does not make use of the knowledge and capabilities they encode and provide. Material forms used for numbers have agency in influencing numerical properties. For example, for the Gooniyandi, Western Australian aborigines, the word three “does not precisely designate ‘three’ (but indicates rather ‘a few’).”17 The Gooniyandi also use the fingers of one hand to indicate the number five. Collecting numbers onto the hand will influence them toward properties implicit in the hand as a material structure and how it is used as a device. Discreteness will emerge because it is difficult for a number that means about three or four to remain fuzzy when it is represented by discrete fingers in sequence. Sequentiality will emerge because it is easier to access the quantities being represented when the hand is used the same way every time; patterned use reduces the demand on 16

Hutchins, 1995, p. 96.

17

McGregor, 1990, p. 149.

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     resources like attention and working memory and improves the information’s accessibility, accuracy, and reliability. And linearity will emerge because it is implicit to the hand’s material structure and its use as a material device, as well as the neurological structure of the hand and the perceptual system for quantity.18 Fundamentally, material agency is revealed by our perceptual limitations: What we can see through numerosity, the subitizable range instantiated by unmodified material forms, is our starting point for numbers; what we cannot see because of the limitations of numerosity motivates our use of material forms to represent and explicate the undifferentiated many that lies beyond about three or four as discrete numbers. Material agency is also found in what the material forms used for representing and manipulating numerical information can and cannot do, capabilities and limitations that drive the mechanism of numerical elaboration. The capabilities of the forms used for numbers interact with our psychological and behavioral capacities in ways that inform numerical organization and structure and infuse the resultant concepts with material properties. Limitations of the material forms motivate us to recruit new devices, while shared capacities of function and form influence what we select. Similarity of function and form are influenced by knowledge and habits acquired with older forms used for numbers; this thread runs through the entire chronology of devices used for numbers. Devices used to represent and manipulate numbers are related by function and form in a way that subsequent devices will build upon previous ones, and previous ones will narrow the range of follow-on devices. In this way, our Western numerals descend from the numerical systems of Rome, Greece, India, Egypt, and Mesopotamia;19 these in turn descend from Neolithic and Palaeolithic forms like pebbles, tallies, and fingers.20

18

19

Penfield & Rasmussen, 1950; Penfield & Jasper, 1954; Harvey et al., 2013, 2015; Patel et al., 2014. 20 Chrisomalis, 2010. Herodotus II:36; Strassler, 2007; Overmann, 2019b.

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    It is possible to approach the idea of material agency in numbers by considering what numbers would be like if we did not use material devices for representing and manipulating them. Numerosity involves the perceptible quantity of material forms, which are agentive in having quantity that we can perceive; the hypothetical removal of material forms to preclude any agentive influence on our perception suggests that ideas of quantity and numbers would no longer have any possibility of existing in the first place. We must therefore ignore the inconvenient circumstance that material forms and their perceptible quantity are the very foundation of numbers, and take our hypothetical case to other material forms found in the typical chronology (Table 7.1). In the absence of any other material forms, we would be unable to transcend the subitizing range, since material forms are critical to realizing discrete quantities from the undifferentiated many that lie beyond it. With language, we would be able to label the few small quantities that we did perceive; this is the condition found in emerging number systems. With the ability to construct ordinal sequences, we might be able to put these labels into a sequence; we might even order them by increasing magnitude – one, two, about three or four – though the hand’s important contribution to structure and stable order would be missing, posing the evolutionary conundrum of how a dedicated neural network for ordinality would emerge in the first place. In the absence of any material means of manipulating and rearranging instantiations of quantity, these verbal numbers would have no more relations between them than are found in ordinal sequences like the days of the week. Calculations and algorithms would not be possible either, as these depend on relations and the ability to manipulate, particularly to numbers of steps that working memory and attention cannot support. Beyond the hypothetical case of what numbers would be like without any material forms, we can also see that which material forms are used and how they are used informs numerical organization and structure. This is yet another agentive influence on their part, as will be seen when different number systems are explored in later chapters.

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    

  MET’s third tenet is enactive signification, the idea that meaning emerges when we engage material forms because of what they are and what we do with them. As we have already discussed how materiality contributes to conceptual meaning, here we want to highlight the difference between material and linguistic signs. The material sign – in this case, any object used to represent number – is visible, tangible, physically manipulable into new forms, and persistent in terms of its physical existence, while the linguistic sign, a spoken word or phrase for a number, is an ephemeral form with none of these qualities. More broadly, any physical object can be a material sign, while any spoken word is a linguistic sign. A material object persists, a quality that helps it anchor and stabilize our concepts of the object itself, as well as its 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. In contrast, a word is ephemeral: Once a word is 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. This fundamental difference in how objects and words mean can be illustrated by thinking about what might happen if we were to give a group of objects to children and ask them either to arrange them or create names for them. Conceivably, those asked to arrange the objects might form ideas about their quantity, ideas prompted by interactions of sorting, ordering, and rearranging. Notions like divisibility, for example, might emerge when the group of objects was split into smaller groups that provide opportunities for noticing the

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    quantity of their elements. This is because things have quantity, quantity is meaningful to numerosity, 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 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. Basically, objects cannot suggest names in language the way they can suggest concepts like four and relations like two plus two. Meaning emerges when and because we interact with material forms. When people first made notches on sticks in the Palaeolithic, they likely did not intend to represent numbers. Rather, the possibility that notches might represent numbers would have emerged through such interactions: first the notches, then the meaning, and then the intention to make notches for the purpose of that meaning. Of course, differentiating between these possibilities is a significant challenge, for they all have the same form. Differentiating an intentional notch from an unintentional mark is not difficult, for unintentional marks are more likely to be haphazard in a way that intentional ones are not; there is also a well-established technique that involves making experimental marks and comparing them to prehistoric ones. The problem is that once the mark becomes intentional, the purpose for which it was made cannot be discerned from the form – a linear mark made for a numerical purpose is indistinguishable from a linear mark made for a nonnumerical purpose, something discussed in later chapters. By drawing its conclusions from a different set of assumptions about the nature of the human mind, MET offers a fresh perspective on the material forms used in numbers, as they are seen as having an

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     essential and primary role in numerical realization and elaboration. The MET perspective also has the potential to critique and change longstanding archaeological assumptions about numbers in the prehistoric record. With that in mind, we will examine those material forms in detail, starting with distributed exemplars and the hand. But first, there is another important aspect of material use in numbers, and that is what happens when it is used in a sustained, collaborative fashion.

 ,   As a primate species, we are highly social, and as the human species, our sociality encompasses a wide range of mechanisms for exchanging information and cooperating toward common ends. Language and behaviors are mechanisms for exchange and cooperation, as is the body of cultural knowledge and norms that we acquire through exposure to social structures and interactions with experienced others. Material culture is another such mechanism, one that also provides us a physical environment of objects with meaning and usability that we learn to recognize and interact with,21 even as it is composed of a multiplicity of discrete objects and intertwined with behaviors and language. But material culture also represents the accumulated cognitive effort of past generations.22 In numbers, material forms structure and anchor concepts realized by past generations, and help ensure their continuity and transmission to future generations across a temporally and spatially distributed socio-cultural fabric composed of material forms, behaviors, language, and concepts. This transmission has made numbers one of the longest-lived cultural systems known for the human species. As a cultural system, numbers represent creativity, but not the type we usually think of. The way that we typically conceive of creativity tends to focus on the individual, the gifted person or genius able to imagine possibilities, experience insights, and devise 21

Malafouris, 2021.

22

Hutchins, 2005.

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    innovations and inventions. Such creative efforts are often acknowledged as taking place within, and thus depending on, social and historical contexts that equip the gifted person with specific resources and thereby shape creative outcomes. What we want to highlight here are the cumulative effects of using materiality. Materiality used in a sustained, collaborative fashion enables the members of a society or social group to create in common. In this process, average individuals participate in quotidian tasks that, taken cumulatively, can ultimately produce complex cultural systems like mathematics and literacy. These mundane interactions involve material forms in ways that change not just the individual, but the social group and the material forms. Such change is incremental – negligible even – and yet in changing, materiality accumulates and distributes collective changes in behaviors and brains, including whatever sparks of innovation are produced by individual creativity. Materiality acts as a medium for sustained, collaborative participation in a shared conceptual system, in which the two-heads-are-better-than-one becomes writ large and small insights, innovations, and functional efficiencies become collected and distributed in ways that preserve and magnify their effects.23 To the individual and a society, this fabric is the cultural environment within which they operate. To history and prehistory, this fabric is the mechanism whereby the conceptual system persists and is incrementally elaborated and extended. Common creativity has another effect: It smooths out the effects of individual variability.24 As individuals, we vary in our psychological, physiological, and behavioral capacities. The material forms we use vary as well, even if they are just another version of the same tool. Much of this variability is subtle and inconsequential, while some of it is dramatic and influential. Some people are just stronger than others, or they have better memory capacities or longer attentional spans, or they grasp concepts more quickly. Some 23

Overmann & Wynn, 2019a; Overmann, 2023a.

24

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Overmann, 2017.

     materials are easier to manipulate and thus more easily modified. These individual differences, taken cumulatively, tend to cancel each other out. In essence, highs counteract lows. Sustained, collaborative use has an aggregating effect that enables the forms and functions of tools to change, yet remain synchronized to average human capacities. This, in turn, means that tool forms and functions will remain accessible to the average individual. Sustained, collaborative use of material forms across thousands and likely tens of thousands of years has bequeathed us numbers that are not just WEIRD but weird, inhabiting, as they do, the continually if incrementally changing end of a trajectory of use and refinement that is tens of thousands of years old. Let us keep this in mind as we look at how things would have started with numerosity and the use of material forms to gain control of it by making it tangible and manipulable.

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

Making Quantity Tangible and Manipulable

Sometime during the Palaeolithic, someone discovered he could use his body to understand and express something he perceived. After a while – perhaps days, perhaps generations, perhaps longer – someone else realized she could use material forms for similar ends, obtaining a wider range of expressive outcomes with greater complexity. Like the body, material forms made the percept visible and tangible, visibility and tangibility made it manipulable, and material forms had greater manipulability and could preserve and accumulate those manipulations to even greater extents than the body could. And whether the body or material forms were used, the behaviors involved were communal: Members of the social group not only performed the behaviors themselves, they also witnessed others performing them. This collaboration took many forms: Some members of the group used their bodies, others the material forms, still more manipulated the material forms into new configurations, and many witnessed and understood what was occurring. Over time, the material forms in question changed in ways that made them better at producing desirable effects, while the effects they produced became more elaborate. This meant that greater amounts of time, practice, and instruction were required to master the material forms in order to produce the desired effects. At the same time, most members of the society understood what was going on and were able to participate in the cultural system. The interesting thing about this narrative is that it applies not just to numbers as the material control of the perceptual experience of quantity, but also to music as a system based on the material control of the perception of sound and geometry as a system based on the material control of the perception of space. Like numbers, music and geometry are cultural systems whose impressive lifespans originated 

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     in the Palaeolithic. Music began as noises made with the voice and the body used percussively; geometry, by using the body to measure spatial extent in physical phenomena. Both would elaborate by recruiting material forms. Music expanded from the body to stalagmites tapped to produce acoustic effects and instruments like flutes, forms attested over the past 50,000 years and thought perhaps as old as 100,000 years or more.1 Geometry, as spatial measurements that supported things like constructing shelters and planting crops, started with body parts like the feet, fingers, palms of the hands, forearms, and the span of the arms held wide open, before expanding to things like the stride, the amount of distance covered by a day’s worth of walking, and ropes and rods with conventionalized lengths. Like concepts of quantity, concepts of spatial extent would become verbally labeled, names reflected today in units like foot, a length of 12 inches originally based on the terminal part of a man’s leg; mile, a unit that originated as the distance covered by a thousand paces; and yard, a rod or stick measuring 3 feet in length. We do not usually think of geometry or music – or numbers, for that matter – as percepts made tractable to manipulation and elaboration by means of bodies and material forms; if anything, we think of geometry as a mathematical science invented by the Greeks, and music in terms of genres like opera, rap, and classic rock. That is, we think of them in terms of historical milestones and contemporary cultural forms. But like numbers and numerosity, geometry and music are perceptual systems made tangible and manipulated, first with bodies and then with material devices. In each case, the perceptual experience preceded the discoveries that the body could be used to understand the perceptual experience and that material forms could be used to produce perceptible effects, and these in turn preceded their subsequent elaboration by means of manipulating the material forms to produce effects of increasing complexity.

1

Atema, 2004; Reznikoff, 2008; Morley, 2014; Díaz-Andreu & Benito, 2015; Fader, 2018.

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      In some ways, spatial measurements are the more apt comparison for numbers because music is not visually perceived, unlike quantity and spatial extent. Both quantity and spatial extent are also initially understood by using the body as a comparison, and they answer similar questions: how many and how much. And both become elaborated as semantically meaningful material representations. Like numbers and fingers, emerging spatial measurements are anchored in the use of the body as a material device for understanding and gaining control of the perceptual experience of spatial extent. For both quantity and spatial extent, material devices – including the body – make the percept tangible and manipulable. Manipulating the percept yields forms, configurations, and behaviors that can be labeled verbally. Here the old story that concepts like numbers are generated mentally and then imposed on passive material forms has been retold. In the revision, percepts like quantity and spatial extent become explicit and explicable as concepts through material tangibility. Such concepts are not merely “unthinkable without the experience of the physical reality”2 of the percept in question; rather, the physical reality is experienced, then understood through comparisons with the physical body, and then manipulated materially in ways that enable the percept to be expressed and elaborated. In this chapter, we will examine how the fingers and other material forms give quantity percepts their initial tangibility and manipulability. As discussed in earlier chapters, the perceptual experience of quantity consists of recognizing, rapidly and unambiguously, small quantities up to about three or four,3 and above that point, discerning quantity differences as larger and smaller if they exceed a threshold of noticeability.4 Essentially, we recognize singles and pairs of material objects, maybe trios and quads, and bigger and smaller in groups. Arguably, these percepts are not yet numbers, which are concepts of quantity shared by sets of objects.5 However,

2 4

3 Renfrew, 2001, p. 98. Jevons, 1871; Kaufman et al., 1949. 5 Brannon, 2006; Piazza, 2011. Russell, 1920.

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     when the quantity of one set is used to express the quantity of another set, it implies the recognition that their quantity is shared.

     When the contemporary ethnographic and linguistic data are considered, especially those of restricted counting systems, several things suggest that material culture is an active and necessary component of numerical cognition. First, initial numbers reflect the perceptual experience of quantity: one and two, sometimes three, occasionally four, and many, often subdivided as big many and small many. Second, analyzable number-words reveal material forms and strategies as their basis: These include objects like the hands used to exemplify quantity, the fingers used to represent and express quantity, and small objects used to manipulate quantity. These material forms make quantity percepts tangible and thus manipulable. Third, the body and material forms are often used without verbal labels, not just for large amounts, but also for small ones; this implies that verbal labeling is a secondary phenomenon to material realization and expression, though nonetheless an important one. In sum, we perceive quantity; we express those percepts with the fingers, material objects, and verbal labels; and we combine material forms to realize new numbers that can occasion verbal labels. When we consider the process whereby we gain control of a visual percept like quantity, it is useful to think about a parallel system. The perception of space (spatial extent, length, or distance) is analogous to magnitude appreciation, the undifferentiated many above the subitizing range, and moreover, it is not burdened by the assumption that the associated concepts originate through language. In the same way that we see many but cannot discern or express “how many” without first making it physically and conceptually discrete in some fashion, we see spatial extent but cannot discern or express “how much” without first making it discrete. In grappling with spatial extent, societies employ the same strategy they do for quantity, which is employing the body and material objects as comparisons

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      that make the percept tangible. Spatial extent becomes tangible as “three feet” when it is the same length as the human foot placed heel to toe in linear succession three times. Similarly, spatial extent becomes tangible as a “yard” when it is the same length as a stick three feet in length. The perception of spatial extent does not change to become discrete, but it can be conceived as three feet or a yard in length when it has been measured and quantified with the body or a stick and can then be imagined or visualized in those terms. Spatial extent made tangible is intelligible, either because every human has an analogue that enables length to be understood in relation to the body, or because spatial extent has been compared to a material object whose length is understood because it is based on the body. In numbers, ethnographic and linguistic data show the use of a similar strategy for understanding the undifferentiated many above the subitizing range: Societies use the body and material objects to make the percept tangible and discrete. A quantity becomes discrete as seven when it is equal to all the fingers on one hand plus two on the other or seven items counted in one-to-one correspondence with the appropriate number of fingers. Societies also arrange material objects in subitizable subgroups to understand nonsubitizable quantities like seven. This strategy is common in early notations, where seven might be expressed as a group of four linear wedges atop a group of three ( Other arrangements were possible (e.g.,

,

,

,

).

), but all of them

employed the same strategy of rearranging a nonsubitizable quantity into subitizable subgroups. The strategy is also found in fingercounting, as for example, seven might be expressed with four fingers on one hand and three on the other, perhaps because the five instantiated by all the fingers on one hand is nonsubitizable.6 These examples reveal similarities and an important difference between spatial extent and quantity. Both use material forms like the feet and hands to realize discrete amounts, and both combine them to realize higher amounts: two feet; both hands. Uniquely, quantity 6

Tempels, 1938.

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     involves discrete percepts – subitizable quantities – that not only act as a starting point, but can also be used as building blocks (for example, seven as four and three, where four and three are subitizable building blocks) or as amounts intermediate between building blocks (for example, phrases like all of the fingers of one hand plus [a subitizable number] of the fingers from the other and all the fingers of both hands minus [a subitizable number] express numbers between five and ten, where the hands are the building blocks and the fingers provide the subitizable amounts needed for the intermediate quantities). The discrete appreciability of the subitizable range provides a starting point in one and two, a mechanism for building both subitizable and nonsubitizable quantities, and a basis for understanding them. The initial numbers are realized by employing the body instrumentally, particularly the fingers; recruiting material objects; and using subitizable quantities and the fingers as building blocks. The material objects recruited generally fall into two categories: common objects with reliable quantity (“distributed exemplars”) and small objects that can be manipulated into numerically appreciable arrays. Within these general parameters, there is a lot of potential variability, and significant actual variability as observed across cultural number systems. Presumably, such variability represents different individuals and societies making different decisions on how to use the hand and which material devices to include. While these decisions can be discerned through the resultant influence on numerical structure and language, their motivation is something we can only guess at. In some cases, there may have been a visual choice, similar to the one in which an image can be seen as either a duck or a rabbit (Fig. 9.1). In the same way, finger-counting systems can take the same pattern – three fingers extended and two bent – and see it as representing either three or two. If the former appears with greater frequency, the latter is not uncommon, and simply represents a different choice between two feasible alternatives. Other potential factors in choosing how the hand is used in counting are biomechanical constraints that tend to limit certain finger movements, cultural beliefs

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     

. . Visual choices. (Left) An image that is either a duck or a rabbit. (Right) A hand configuration that is either three (extended fingers) or two (folded fingers). Both images are in the public domain.

that certain finger patterns are to be avoided, and cultural forms that may inform specific choices of finger movements and combinations. Most significantly, material objects do not have start or stop points.7 Just as there is no internal impetus to use the hand in counting in any particular way, there is nothing about the hand itself as a material form that mandates that it be used for counting in a specific fashion. As previously mentioned, the Mundurukú use the arms to express the number two as part of a compound expression for three,8 an example of using common objects with reliable quantity (“distributed exemplars”), as well as using subitizable building blocks. Similarly, the Bakaïrí used kernels and the fingers to visualize the number three,9 recruiting and manipulating small objects, using the fingers instrumentally to anchor and express the quantity concepts, and using subitizable building blocks (two and one) to form a compound (two and one, which is three). We will discuss these strategies more fully below, particularly in regard to their influence on verbal labels for numbers. 7 9

8 Malafouris, 2013. Pica & Lecomte, 2008; Rooryck et al., 2017. Von den Steinen, 1894a.

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    

   Western people tend not to think of the body in instrumental terms, as it has long been far more common in the Western cultural tradition to use elaborated cultural systems like measurements, maps, and symbolic notations to mediate percepts of spatial extent and quantity. Using the body to estimate aspects of the world is a fundamental strategy employed by many species, not just the human one. The prevalence of this strategy likely reflects the evolutionary advantages of being able to judge whether something is big enough or numerous enough to be a threat or an advantage. Thus, it is not surprising to find human societies using the body as the comparison for estimating visual percepts like spatial extent (size in feet, etc.), direction (e.g., front, back, side, right, left, up, and down), and quantity (as many as the fingers on my hand). As a material device for comparing and counting quantity, the hand has some undeniably peculiar properties that set it apart from traditionally material devices like tallies and tokens. For one thing, it is intimately connected to the brain through the nervous system; for another, the region of the brain that “knows” the fingers is neurally integrated with the part that perceives quantity.10 In addition, the hands are almost always within sight, and this is important, for as previously noted, people blind from birth do not count on their fingers.11 These properties make the hand both an actor and an instrument, both able to act and to be acted upon. This is an ability no other species appears to share, since paws or wings are used to act and sense but not as objects themselves. Not even the chimpanzee, today the living species most closely related to humans, uses its fingers instrumentally in counting. This ability to use the hand instrumentally makes it a powerful bridge between the internal and external domains of experience, a quality that enables it to

10

11

Marinthe et al., 2001; Roux et al., 2003; Penner-Wilger et al., 2007; Reeve & Humberstone, 2011. Crollen et al., 2011.

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      integrate the psychological, behavioral, and material dimensions of numeracy.12 As a material device for counting, the hand is extremely perishable. Finger-counting and finger-montring (the use of the fingers for numerical display described in Chapter 4) are ephemeral gestures, and sooner or later – and probably sooner – the hand will be needed for something other than representing quantity. Most people do not walk around with their hand showing the number three, for example, not just because they cannot remember to do it all day long, but also because as long as they are doing it, their hand cannot easily be used for anything else. And a material device, the hand is also not very manipulable. Sure, the fingers are easily moved to accumulate, but it is harder to divide them into different groups, and as for subtraction, the fact that the fingers remain connected to the body seems to make them not as subtractable as, say, removing a pebble from a pile of them does for that material form. And third, the hand as used in emerging number systems does not have a large capacity for numbers, even if the actual amount varies by how the hand is used: If the digits are used without the thumb, then the fingers of one hand typically count to four; if the segments are used and the thumb included, then the fingers of one hand can count to fourteen. Other variations were noted in Chapter 4. The hand’s structural influence on numbers is related to how it is used as a material device, as well as what it is as a device. Look at the hand and think about the perceptual experience of quantity: one and two, about three or four, and many. The hand enables expression of the idea as many as the fingers on my hand, an initial step into the undifferentiated many of magnitude appreciation. Five cannot be appreciated like one through three, since five is not subitizable, but the hand functions as a visual reference that can express five as this many. Now, look at the hand with the ideas of one and two with the little and ring fingers, and the idea of as many as the fingers on one 12

Gallagher, 2013; Malafouris, 2013.

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     hand with all five fingers. What happens to about three or four? This fuzzy number is left to inhabit the index and middle fingers, which fall between the little and ring fingers and the thumb. Here the form of the hand plausibly influences the idea of about three or four toward discreteness as three and four. In the process, the numbers three and four are also ordinally sequenced, since they come after one and two and before five. The hand as a physically coherent device consisting of five fingers provides an integrity that is critical in numbers. Consider the use of distributed exemplars: the eyes or the arms for two; a tripod, a pronged arrow, or three-clawed bird toes for three; an animal coat with four spots for four. These disparate and physically separated items do not constitute a coherent physical structure for representing quantity like the hand does. The discontinuity inherent in perceiving quantity across the material environment but not collecting it onto a single device would likely preclude the realization of higher numbers, something for which a material device is essential; it would also preclude concepts of the subitizable numbers from becoming discrete, linear, and stably ordered, properties they acquire from moving sequentially across the fingers as elements of a coherent, contiguous physical structure.13 Human numbers must differ significantly in this regard from whatever it is that other species have for number concepts, as no other species counts on its fingers or digital pads, not even the chimpanzee. While finger-counting patterns vary cross-culturally,14 all known variants involve choosing some feature to start and stop counting (e.g., an outside finger or finger segment) and then proceeding sequentially in some fashion from the starting feature to the ending one.15 In other words, social groups tend to do the same thing – count with their fingers – in different ways. This suggests that patterns may involve but are not determined by the brain’s topographic

13 15

Gelman & Gallistel, 1978. Overmann, 2014.

14

Huylebrouck, 1997; Domahs et al., 2010.

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      sensorimotor structure;16 instead, they are mediated by the material features of the hand, social choice, and cultural exposure. Fingercounting patterns also become habitual because they are done the same way every time. The reasons for this become apparent when the alternatives are considered: Using the fingers randomly would be less consistent and reliable, while using the fingers nonsequentially would be biomechanically awkward and less distinguishable visually. By comparison, starting and ending with the same features and proceeding between them in the same fashion reduce demands on memory and attention, facilitate biomechanical production, and improve informational consistency and reliability, matters that are further improved by behavioral automaticity, becoming able to perform behaviors without really thinking about them; automaticity is gained by repeating the behavior to the point where its performance no longer requires sustained, dedicated attention. The hands and feet are also naturally grouped into fives, tens, and twenties; this is a common factor in how the hand is used, but not an invariant one. This natural grouping tends to make five the capacity of a single hand and ten the capacity of both hands. Essentially, capacity is governed by what the hands are and how they are used as a material structure, and there are multiple strategies for extending their capacity, as we will see in the next section. The hands do not specify what they count, so that information must be maintained elsewhere, which typically means – absent a method of marking or recording – that it is kept in memory or in context. In some languages, the kind of object counted is marked in language, perhaps by adding another syllable or by changing an internal vowel, so that the numbers used with, for example, sugar cane are different from those used with coconuts, which are different still from those used with fish or things that are not any of the above. This kind of counting – different ways of counting different types of objects, or objectspecified counting – is found all over the world, and was the type of 16

Harvey et al., 2013.

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     number system used by the ancient Sumerians; we will look at a contemporary example from Polynesia in Chapter 12.

         The Mundurukú of Amazonian Brazil count from one to about four.17 Above about four, they use the idea of a handful18 (Fig. 9.2, left), grounding numerical concepts in the use and manuovisual experience of the hand. This example involves the hand more for what it can grasp than for the quantity of its fingers, potentially collapsing the distinction between how many (quantity) and how much (spatial extent). The hand can also be used without individuating the fingers, creating phrases whose meaning is typically given as many as the fingers on my hand (Fig. 9.2, middle). An example is found in the Yamana of Tierra del Fuego, whose word for five was yëkýli yéš, meaning one hand, while ten was kompéi yéš, two hands; the only other Yamana numbers were terms for one, two, and three.19 The fingers can also be individually enumerated (Fig. 9.2, right), as is the case for the Amazonian Desana, whose terms for five and ten – yuhuru mõhõtõ (one hand) and pẽmõhõtõ (two hands) – are accompanied by companion terms for the remaining fingers; for example, the term for six is yuhuru mõhõtõ yuhuru nĩãɾã ̃ , [all the fingers of] one hand [and] one [finger] from [the] other hand.20 What these examples show, particularly Fig. 9.2, middle and Fig. 9.2, right, is the use of the hand to represent five, the first nonsubitizable quantity. This, along with gestures like finger-counting and finger-montring, involve the hand in counting, beginning the process whereby number concepts acquire properties like linearity, stable order, and discreteness from the hand and the subsequent material forms used to represent and manipulate them. Numbers higher than five or ten can be counted with the feet (Fig. 9.3, left). For example, the Luiseño of California counted “chiefly 17 20

Rooryck et al., 2017. Silva, 2012, p. 183.

18

Pica and Lecomte, 2008.

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19

Gusinde, 1931, p. 1838.

     

. . Using the hand and fingers in restricted counting. (Left) Handful; (middle) without individuating the fingers; (right) with the fingers individuated. Images of hands are from the public domain. Adapted from an image published in Overmann (2021d, Fig. 1a–c, p. 3).

. . Using more positions on the same body. (Left) Adding the feet; (right) adding other body parts. Diagram of Oksapmin body-counting adapted from Saxe (2012, Fig. 12, p. 46). Images of hands and feet are from the public domain. Adapted from an image published in Overmann (2021d, Fig. 1d–e, p. 3).

by means of the fingers and toes.”21 This method characteristically creates groupings by fives and twenties: ten, all my-hand finished; fifteen, all my-hand finished and one my-foot; twenty, another finished my-foot the-side.22 Numbers higher than ten can also be counted by continuing from the hand to other parts of the same body, using it like an ungrouped tally. For example, Oksapmin bodycounting (Fig. 9.3, right) counts to 27 by sequentially moving from

21

Sparkman, 1905, p. 657.

22

Dixon and Kroeber, 1907, p. 689.

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     the fingers of the right hand up the right arm, across the head, and down the left arm before finishing with the left fingers.23 The hands of the same person, with or without the feet, can also be reused to count additional cycles of ten or twenty. This requires keeping track of the number of cycles in some fashion. In Papua New Guinea, “the Karam-speaking people, who live near Aiome in the mountainous inland of the Madang District, usually count on their bodies to a total of twenty-three on the first time across, and then may turn back to count to twenty-two each time they cross from the little finger of one hand to the other (they do not count the last finger of one circuit when they begin the next; it is counted only once).”24 For the Iqwaye of Papua New Guinea, after twenty is reached, the hand is used to accumulate multiples of twenty; after 400 is reached, man (“as many persons as one person with all their hands and legs” or twenty times twenty) is used as a unit to accumulate multiples of 400.25 Formations like the Luiseño self-reference suggest that the person counting might be keeping track of the number of cycles in some fashion, likely the fingers (Fig. 9.4, left). Luiseño numbers also show that while some possibilities are mutually exclusive – counting cannot be both grouped and ungrouped, for example – others are not, since self-reference (my hand) can be combined with counting the hands alone or both the hands and feet. Particular fingers can also be used for keeping track of the tens, hundreds, and thousands (Fig. 9.4, right), as was the case with the finger-counting system recorded by the Benedictine monk known as the Venerable Bede in 725 CE.26 In Bede’s system, units were displayed with the little, ring, and middle fingers of the left hand, tens with the left index finger and thumb. The right index finger and thumb displayed the hundreds, while the right little, ring, and middle fingers tracked the thousands. Numbers in the range 10,000–90,000 were indicated by touching various parts of the body with left hand,

23 26

24 Saxe, 2012. Wolfers, 1971, p. 82. Bede, 1999; also see Nishiyama, 2013.

25

Mimica, 1988.

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     

. . Reusing the fingers of the same person. (Left) Remembering cycles; (right) using the fingers to track cycles. Images of hands are from the public domain. Adapted from an image published in Overmann (2021d, Fig. 1f–g, p. 3).

100,000–900,000 by touching with the right hand, while 1,000,000 was indicated by clasping the hands together with interlocked fingers.27 Keep in mind that Bede’s system represented decimal numbers that counted up to a million, a highly elaborated system that differs significantly from the kinds of emerging systems that are the focus of this chapter. Finally, multiple people can also collaborate in counting. One technique adds the hands (and toes) of additional people to extend the range of counting serially (Fig. 9.5, top). In Papua New Guinea, someone needing to count beyond twenty, having exhausted his own fingers and toes, uses the fingers and hands of someone else.28 The Inuit, a people of the Arctic regions of Canada, Greenland, and Alaska, counted in this manner as well, with twenty-one, inûp áipagssâne atausek, meaning one [finger] on the second man; thirty-eight, inûp áipagssâne arfinek pingasut, three toes on the second man’s second foot; and forty, inûp áipagssâ nâvdlugo, the whole [digits] of the second man.29 Another example comes from the Zande (or Azande)

27 29

Richardson, 1916, pp. 8–9. Nansen, 1893, pp. 194–195.

28

Zarbaliev, 2015, citing Miklouho-Maclay, 1951.

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    

. . Recruiting additional bodies. Key: (Top) Sequential collaboration (in a vigesimal example) extends the range of the tally by adding the digits of more people, each of whom counts their digits; this technique extends counting but does not produce positional structure. (Bottom) Positional collaboration (in a decimal example) divides responsibilities between people, each of whom counts from one to ten but weighs them as units, tens, hundreds, and so on; this technique both extends counting and produces positional structure. Adapted from an image published in Overmann (2021d, Fig. 1h–i, p. 3).

in Central Africa, today the northeastern part of the Democratic Republic of the Congo, where counting beyond twenty meant “another person is requisitioned, and the reckoner proceeds to count his fingers and toes, and so on.”30 Another collaborative technique divides the responsibility for keeping track of the units, tens, and hundreds between multiple people (Fig. 9.5, bottom). This behavior has been documented in Africa, Oceania, and Papua New Guinea.31 In this type of counting, the first man “counts the units on his fingers by raising one finger after the other and pointing out the object counted or, if possible, touching it. The second man raises a finger . . . for every ten, as soon as it is completed. The third man counts the hundreds.”32 Similarly, 30 31 32

Gore, 1926, p. 42. Hazlewood, 1850; Schrumpf, 1862; Collocott, 1927; Lancy, 1978. Cajori, 1899, p. 35.

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      an early Fijian dictionary defined bunu-ca as a verb that meant “to tally, or count the number of tens while another is counting the units.”33 In Papua New Guinean body-counting, “if a complete revolution is made without exhausting the collection, the counter asks associates to stand in place of each revolution as he goes on counting. The final tally is then expressed as so many men and some remainder.”34 And in Polynesia,

[When the man counting] reaches ten a second man . . . calls “one”: the first man continues counting to twenty, when the second man shouts “two,” and so on up to one hundred, when the second man calls ”ten,” and a third man gets up . . . and shouts “one.” So they go on, the first man actually counting, the second keeping count of the tens, and the third of the hundreds. Should the number of [items being counted] go into the thousands a fourth man would join in. (Collocott, 1927, p. 26)

All of these methods influence numerical properties like extent and organization. Restricted number systems involve the hand for what it can grasp (Fig. 9.2, left), as an uncounted, visually appreciated standard (Fig. 9.2, middle), and with the digits individualized to represent discrete numbers (Fig. 9.2, right). Using more positions on the same body extends the upper limit of counting, while either influencing numbers toward grouping by five and twenty (Fig. 9.3, left) or incrementing without grouping (Fig. 9.3, right). Reusing the fingers of the same person either requires remembering the decades as counting continues (Fig. 9.4, left) or using the fingers to track them (Fig. 9.4, right). Finally, additional bodies can be recruited, either extending counting by adding additional cycles in sequence (Fig. 9.5, top) or keeping track of units, tens, and hundreds in a positional manner (Fig. 9.5, bottom).

33

Hazlewood, 1850, p. 20.

34

Lancy, 1978, p. 6.

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    

 ’     Once upon a time, one of the Brothers Grimm noted that the fingers were the source of names for the numbers.35 Yes, those Brothers Grimm, Jacob and Wilhelm, famous for the folk tales they collected and published: “Cinderella,” “Hansel and Gretel,” “Rapunzel,” “Rumpelstiltskin,” “Sleeping Beauty,” and “Snow White.” Beyond their interest in folklore, both brothers were accomplished academics, and the elder, Jacob, was a well-regarded linguist. Jacob formulated what would be called “Grimm’s dictum”:36

All words for numbers come from the fingers of the hands, as is still the case now with peoples who use lively play of signs, especially Italians, who tend to extend their fingers to count. (Grimm, 1868, p. 167, as translated)

In actuality, the use of the fingers in naming numbers does not appear to be as straightforward as Grimm thought. As would be observed nearly forty years later, “scarcely any language bases the numeral-words One to Four on finger-names. Presumably this is because the earlier numerals had been formed before a step upward in the art of notation was taken, through the idea being conceived of using the fingers as an aid in that process.”37 Certainly, across languages and cultures, the names for the subitizable quantities one through four, particularly the terms for one and two, are often unanalyzable. While this property has implications for numerical age and usage, as discussed in previous chapters, it does not mean that the fingers were necessarily uninvolved. Indeed, analyzable terms “often express their affiliation with the body,”38 and using the body to make sense of the world is a typical strategy that is not unique to the human species. This suggests it is quite possible that many unanalyzable

35 38

Grimm, 1868. Dorais, 2010.

36

Trumbull, 1874, p. 19.

37

Bagge, 1906, p. 260.

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      forms for the numbers one through four were originally based on the fingers, but have become etymologically opaque through frequent usage and the passage of time. Analyzable numbers and ethnographic observations often show the fingers as the basis for realizing and then naming the initial numbers, typically in ways that suggest that the use of the fingers precedes verbal labeling. An example is found in the Bergdama (or Damara), a semi-nomadic people of central Namibia. The Bergdama counted the numbers one to five on the fingers of the left hand; they used the identical phrases to count the numbers six through ten on the right hand, with the use of the left or right hand indicating the intended register:

6¼kari gaoneb 6¼kari gaoneb !gãb //aiga mãbeb //naibeb gei khoi khaobes

(left small finger; “little chief”) (left ring finger; “brother of the little chief”) (left middle finger; “the one standing in the middle”) (left index finger; “the pointer”) (left thumb; “tall man standing behind”)

=1 =2 =3 =4 =5

[Note: 6¼, //, and ! represent different click sounds.] (Vedder, 1923, pp. 165–166, as translated)

Combining words and fingers often appears in African number systems as recorded by Western researchers in the nineteenth century. For example, in eastern Namibia, the term neba hawu was used for seven, eight, nine, and ten, a verbal indeterminacy that required the requisite fingers be displayed to express the intended number.39 The spoken phrase served mainly to draw attention to the hand as the fingers were displayed. Words might also be entirely omitted in favor of gesture, as noted by the naturalist Henry Lichtenstein during his

39

Olderogge, 1982.

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     travels in South Africa. Of the Xhosa, he wrote, “Sometimes . . . the numeral is not mentioned; the action of raising the fingers only is used”; of the Tswana, “The numbers are commonly expressed . . . by fingers held up, so that the word is rarely spoken: many are even unacquainted with these numerals, and never employ any thing but the sign.”40 If not the fingers, terms for one and two might describe singles and pairs as alone and together, terms with the sense of comparing the subitizable quantity of singles and pairs. Terms for two, three, and four might also be compounded from subitizable numbers, as in a term for two that means one and one, a term for three that means two and one or one and two, and a term for four that means two and two, three and one, one and three, one from five, or five less one. Such compounds imply the involvement of the fingers and/or small, manipulable objects. Interestingly, our sensitivity to numerical quantity improves for objects that are sized to be grasped,41 something that is plausibly a factor in our use of small, manipulable objects for representing quantity. As noted in Chapter 4, the fingers were involved both as an exemplar of quantity and in manipulating small objects when the Bakaïrí used kernels and fingers to visualize the conjoining of the terms for two (aháge) and one (tokále) to produce a term for three (aháge tokále). Compounds for two and three are particularly interesting because both are subitizable, and yet, compounded terms for both are not uncommon, particularly for three. This suggests that the appreciability of subitizable quantities does not necessarily determine their realization as numbers; rather, their realization is accomplished by manipulating small objects in ways that makes their quantity conceptually explicit. Functionally, the subitizability of both addends and sum might facilitate the understanding of the compounding operation in its entirety, perhaps a necessary precursor to the further compounding that produces sums exceeding the subitizing range. 40

Lichtenstein, 1812, Appendix.

41

Ranzini et al., 2011.

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      Terms for three and four might also be named after specific fingers, properties of specific fingers, or the place that specific fingers occupy in a sequence of fingers: for example, little finger (a specific finger), weak nail (i.e., a property that designates the thumb), and the one standing in the middle (the finger in the middle, regardless of which side of the hand is used to initiate counting when fingers are counted from one side of the hand to the other).42 Such labels imply that the hand is used to represent the numbers in question, occasioning and influencing the form of their verbal labels. Other objects are sometimes leveraged to exemplify small quantities. These include the arms (Mundurukú), eyes (Hup), ears (Chinese), and deer footprints (Xerénte) used to exemplify two; rubber seed (Nadahup family), bird footprint (Xerénte), jar support (Yucuna), and pronged fishing arrow (Hixcaranya) used to exemplify three; and the toes of an emu and the spotted coat of an animal (Abipónes) used to exemplify four and five, presumably because these items occurred reliably in the indicated quantities.43 In this regard, the hands are the quintessential distributed exemplar, distributed because hands appear wherever people are and exemplar because five fingers are the norm. The fingers not only occasion and influence the initial verbal forms for signaling and naming the fingers, they also support the truncation and shortening of lengthy phrases to shorter forms. An example is found in the part of the Democratic Republic of the Congo once known as Zaire, where a term that originally meant there is missing one out of ten, pabula kimɔ kia likumi, can be shortened to pabula, there is missing.44 The numerical meaning of the verbal term is supported during its truncation by the accompanying display of fingers, suggesting that verbal numbers are closely related to the fingers not just in their emergence as names for the fingers used in counting, but also in their achieving the shorter, less analyzable forms 42 43

44

Olderogge, 1982. Dobrizhoffer, 1822; Gow, 1884; Pica & Lecomte, 2008; Epps et al., 2012; Rooryck et al., 2017. Tempels, 1938, p. 52.

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     that are more convenient and less demanding of cognitive resources like attention and working memory. This material role in anchoring meaning recalls the differences between a word for an object and the object itself, the linguistic and material signs discussed in Chapter 8: Words are ephemeral and their meaning is conventional, while the material form persists and means what it is.45 The initial numbers are realized by using the fingers and small objects to make quantity tangible and manipulable; five, ten, and twenty are realized by using the fingers, toes, remembered or tracked cycles, and extensions that include other bodies; and the gap or intervening numbers are realized through compounding, by either adding subitizable quantities to building blocks or subtracting from them. The range of potential combinations, along with differences of longevity and usage that influence analyzability, account for the variety of naming conventions found in the linguistic data. Crosslinguistically, verbal constructions for numbers fall into three categories. The subitizable numbers are often highly irregular atoms, something that may reflect an origin in distributed exemplars, while the first few decades (or twenties, etc.) above them are irregular compounds, something that may reflect an origin in idiosyncratic compounding. In English, these ranges would correspond to the numbers one through ten and eleven through nineteen; in Sumerian (Fig. 5.3), to the numbers one through five and six through fifty. Such names reflect the use of material forms (fingers, objects), the absence of generating names by means of lexical rules, and the fact that emerging number systems have less structure than do more elaborated ones. Numbers higher than irregular compounds are generated by means of lexical rules, which makes their constructions highly regular.46 As mentioned in Chapter 5, lexicalization presupposes the availability of atoms, counting to higher numbers, and frequent use of numbers; some ideas of how lexicalization might emerge are given in Chapter 12. 45

Malafouris, 2013.

46

Greenberg, 1978.

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     

-  How long have we been counting on our fingers? If the hand is the first material device we use for counting and numbers are tens of thousands of years old or even older, then finger-counting surely has an impressive prehistory. This is not generally something we can see archaeologically, since gestures like finger-counting do not typically leave material traces behind. But in 2006, a French researcher named André Rouillon noticed something unusual about the 25,000-year-old hand stencils in Cosquer Cave,47 a site located on the southern coast of France whose entrance is now some 120 feet underwater. Cosquer contains over sixty hand stencils, about two-thirds of which were made with a black substance, with the other third in red (Fig. 9.6).48 Cosquer contains one of the largest collections of hand stencils in all Europe, though comparable numbers of hand stencils have also been found at Gargas Cave, France (27,000 years old), and El Castillo, Spain (40,000 years old).49 The Cosquer stencils are negative, meaning that they were formed by placing the hand on the cave wall while blowing or dabbing paint around it. A positive stencil, in comparison, would be formed by dipping the hand in paint and then touching it to the wall. Some of the fingers of the Cosquer stencils are shorter than is normal for the human hand. Early researchers speculated that the shortened digits represented fingers lost through frostbite, hunting accidents, or mutilation rituals;50 debate over the reason for the shortened digits remains ongoing.51 When finger-counting involves extending or folding the fingers of one hand, the numbers one through five require five different patterns. But as Rouillon pointed out, it is possible to form thirtytwo different patterns by extending or folding the five fingers of a single hand; that is, from no fingers extended to all five extended,

47 49 50 51

48 Rouillon, 2006. Clottes & Courtin, 1994; Clottes et al., 2005. Clottes et al., 1992; Pike et al., 2012. Luquet, 1926, 1938; Leroi-Gourhan, 1967. E.g., McCauley et al., 2018; Alonso, 2019.

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    

. .. Two hand stencils from Cosquer Cave. (Left) A black stencil with all five fingers extended. (Right) A red stencil with only the thumb extended. Images © Jean Clottes and used with permission.

thirty-two patterns are possible.52 Of the thirty-two possible patterns, only five are found at Cosquer, the ones used in sequentially counting from the thumb to little finger by extending the digits. This is a nonrandom distribution, and because it is consistent with a common way of counting on the hands, it suggests finger-counting.53 Similarly, at Gargas, five of the ten hand-stencil patterns are the same as those found at Cosquer; the other five patterns at Gargas are thought to have possibly been hunting signals.54 On the other hand (no pun intended), which fingers were truly short or long can also be difficult to discern, as paint is degraded by moisture and other natural conditions, and not all of the hand stencils at Cosquer were included in Rouillon’s study. The hand stencils also 52 54

53 Rouillon, 2006. Overmann, 2014. Leroi-Gourhan, 1967; see analysis in Overmann, 2014.

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      appear to have been made by individuals who varied in gender and age;55 the stencils also differ in their color, which hand was used, how they are oriented, which images were painted nearby, their own repetition as groups of signs, and their placement within the cave. Numbers and hunting signals are not the only interpretation: The signs have also been interpreted as a form of sign language.56 Even if the Cosquer hand stencils had a numerical intent, their other characteristics have potential implications for nonnumerical social meanings and purposes. If the hand stencils were meant to be numbers, they are not unambiguous as such to our eyes, so they have not yet yielded all their secrets about their meanings and purposes within Upper Palaeolithic life. In short, if one of the finger-patterns meant three, we still do not understand the purpose of repeatedly stenciling three in red on a particular wall of the cave. As was true of the 30,000-year-old Sunghir burial in Russia, the complex parietal art at Cosquer Cave, of which the hand stencils form only a part, suggests the kind of socio-material complexity predicted to motivate numerical elaboration.57 As previously noted, the presence of the conditions that motivate numerical elaboration does not entail that numerical elaboration had actually occurred. Certainly, the hand stencils at Cosquer suggest counting no higher than five: The infrequent use of the stencil for four, along with the absence of any combinations suggesting numbers higher than five, suggest a number system that counted the subitizable numbers one to three and added five as all the fingers on one hand.58 The difficulties inherent in determining whether or not Palaeolithic hand stencils had numerical meaning is unfortunately just the beginning of the challenges faced by archaeologists trying to find numbers in the material record. We will see this again, and in greater detail, when we look at tallies and other one-dimensional (accumulating) forms in the next chapters.

55 57

56 Snow, 2013. Etxepare & Irurtzun, 2021. 58 Owens & Hayden, 1997; Hayden, 2021 Overmann, 2014.

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 Tallies and Other Devices That Accumulate

Consider the humble tally. Whether it is made of notched wood, knotted string, a torn leaf, strung beads, loose pebbles, marks painted on the body or inscribed on the ground, the fingers, the fingers and toes, or the fingers plus other body parts, a tally is a simple device, as material forms go, one that requires few resources to learn or invent from scratch. But because it is a material form that is not a part of the body, the tally represents an extremely powerful mechanism – the ability of the material form to accumulate and distribute cognitive effort – that for numbers begins with the tally and continues today with calculators and computers. If the tally is easy for a novice to understand, use, make, and invent, a device like the computer is not, even for an expert. This is because at some point, the amount of cognitive effort needed exceeds what a single individual, or even an entire generation of people, can manage on its own. Material devices also have a capacity for manipulability and morphological change that far exceeds what bodies and behaviors are capable of; they are also public and shareable in ways that bodies and behaviors are not. The tally thus represents a significant step in harnessing the agency of material forms toward numerical purposes. Why this is so is easy to see when the tally is compared to finger-counting, a social behavior that new individuals learn to recreate with good fidelity. However much the hand is used for fingercounting, its form does not change in response to the behavioral and psychological changes that users acquire through practice. Yes, the hand can and will become more skilled under conditions of sustained use, in things like hand-eye coordination or, in the case of fingercounting, the ability to produce certain finger-patterns, but it will not acquire additional fingers, nor will the fingers change in their 

https://doi.org/10.1017/9781009361262.012 Published online by Cambridge University Press

       length or spacing to form numerical codes. Only material forms are capable of such plasticity, with change in material form responding to the use of the device in a way that emphasizes and intensifies its usable features.1 Morphological plasticity is what enables a material form to accumulate incremental change in the behaviors and brains of its users, change that is distributed to new individuals and generations when they learn to use the device. In terms of function, the tally and other devices that accumulate are similar to the fingers: Sticks or bones with notches, strings with knots, leaves with tears, marks on surfaces, beads on strings, and loose stones all accumulate like the fingers do. And as is true of the fingers, each element (notch, knot, tear, mark, bead, stone) represents one of whatever is being enumerated in one-to-one correspondence; unlike the fingers, which are individuated in terms of their size, length, and location on the hand, the elements of a tally are identical in their form and function to the other tally elements (e.g., | | | | | | | | | | | |).2 Elements that are parallel linear marks (e.g., notches, tears, marks) have a form that is similar in some respects to the fingers. Parallel linearity is unlikely to be based solely on the resemblance. This is because there are reasons like visual distinguishability and the relative ease of making straight incisions in wood (rather than making, for example, circles) that would likely influence marks to take the form of parallel lines; the linear form of tally notches would act to reinforce the linear form of the fingers, likely the reason that the same form recurs in notations. Once notches have been made (or loose objects like stones have been collected), accumulated elements also persist in a way that fingers cannot when used as a device for representing numbers, and they have the potential for greater capacity than what the fingers can provide. These capabilities and limitations are the basis for the recruitment of these material forms: Tallies and other devices that accumulate are a type of material form whose form and function are systematized by what the fingers can and cannot do. 1

Overmann, 2021f.

2

Schlaudt, 2020.

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     Tallies and other surface marks are limited in their manipulability, to such a degree that it is difficult to move or remove a mark once it has been made. Of course, this depends entirely upon whatever substance the marks are made on: Notches incised into wood are more difficult to move or remove than are stripes of mud that have been painted onto a washable surface.3 Pebbles and other loose forms (e.g., beans, kernels of corn) have a high degree of manipulability, though this also means that they lack integrity of form. It is possible to drop, scatter, and lose track of loose objects in a way that is difficult with a form like the tally, whose material integrity means that it is difficult to lose the individual marks, though it is obviously possible to lose the entire device. Given an increased social need for calculating and recording, these characteristics will motivate the recruitment of material forms that provide the needed capacity for manipulability or integrity, and the material forms recruited for this purpose will be systematized by the tally’s capabilities and limitations, in the same way the tally was previously systematized by the capabilities and limitations of the fingers. Cross-cultural data on numerical traditions show that systems using both a form like the tally for recording (representation) and a form like loose pebbles for calculating (manipulation) are fairly common. There are also number systems that only tally (accumulate and represent), as for example, digit-tally systems like that of the Desana, in which the fingers and toes are the only form of tally used, and the body-counting systems of Papua New Guinea, in which a sequence of body positions are the only form of tally used. Tallyonly systems suggest that accumulating quantity tends to emerge before manipulating quantity does. Systems that only tally, and ones that both tally and manipulate, share the ability to accumulate; accumulation acts as a central organizing principle for the resultant numbers, which, when they add grouping, will increase in a predictably linear way (e.g., by multiples of ten or twenty). In contrast, 3

Morgan, 1852.

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       numerical traditions in which manipulation acts as the central organizing principle are rare. They do not increase in a predictably linear way, but instead combine and mix numbers in a nonlinear fashion. As we will discover in Chapter 12, some number systems in West Africa may be examples of this type. The issues that we will consider in this chapter are these: First, we will look at what a tally means in terms of numerical organization and structure, using as an example the body-counting of the Oksapmin of Papua New Guinea.4 Both the body as used as a material form for counting and the verbal names for the different body-parts comprise an ordinal sequence, and when both the material and linguistic forms have the same properties, they are likely to reinforce each other. This means that the number system is likely to be highly stable and persist over long periods of time, especially in the absence of any emergent social need for anything except ordinal counting. Next, we will examine what is involved in transitioning numbers from the body to a material form like the tally. For this, we will look at the number system of the Desana, an Amazonian people who count to twenty on their fingers and toes5 but do not tally with material forms. This in turn will let us ask what might be involved in transitioning from the use of the body as a tally to the use of a material form like a notched stick. Finally, we will see that tallies are used to count to numbers well beyond the range of the available numerical vocabulary.6 This will provide insight into the role of material forms in extending numerical extent. It will also enable us to ask how and when the individuated forms used for counting – each finger used in digit-tallying, each notch inscribed on a stick to represent a particular item – become deindividuated or “tokenized” so that each is just another element or “token” added to the sequence instead of an individual joining the sequence.7 Throughout the discussion that follows, the term “tally” is used to express both the material form of a stick incised with notches and

4 6

5 Lean, 1992; Saxe, 2012; Owens, 2018. Miller, 1999; Silva, 2012. 7 Morgan, 1852; Blake et al., 1998. Schlaudt, 2020.

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     the broader category of devices that accumulate. The use of the term “tally” to denote the broader category means that an argument parallel to one developed for the stick tally is usually applicable to any material form that accumulates, albeit that for devices which accumulate, fixed forms are used to represent (record), loose forms to manipulate (calculate). The points important to keep in mind are these: First, the properties of any particular material substance will influence how it will be used in representing or manipulating numbers. Second, both the properties of the material substance and the way in which a particular material form is used will influence numerical outcomes, particularly in how numbers are conceptualized. Simply, in emerging from the perceptual experience of quantity, numbers are realized and elaborated from that perceptual starting point through the use of material devices. This means that the specific material devices used to represent and manipulate them will directly inform how they are conceived.

   When societies need to count to higher numbers than what the fingers provide or represent them for any period longer than what is possible with the fingers, be that minutes, days, years, or longer, they recruit a material form that does what the fingers do – accumulate – but which has greater capacity and persistence. Not all solutions solve both problems, as for example, collaborative finger-counting, in which the fingers (and perhaps toes) of a second person extend the count made with the first person’s digits (Fig. 9.5, top), increases capacity but not persistence. Any improvements in persistence are also relative, as for example, a leaf is more perishable than string, which in turn is more perishable than wood, which in turn is more perishable than bone. Sequential accumulation is inherently linear. An interesting question is why these different forms accumulate in a linear fashion, when there may be little or no need for them to do so in virtue of the material substances they are. This is not true only for pebbles, since

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       the marks on a tally can be made anywhere on the piece of wood. However, they tend not to be, in favor of proceeding from one end of the device to the other. In fact, archaeologists interpret haphazard placement as indicating marks made unintentionally, perhaps as created inadvertently during a process like butchery, and unintentionality is seen as the opposite of marks created with some social purpose in mind, the category to which marks made to represent numbers belong. For numbers in particular, the way our brains are neurologically “wired” may influence numbers toward linearity; so too does the physical arrangement of the hand as it is used for counting, as for example, moving from the little finger across the hand to the thumb. This linearity becomes part of how we conceptualize numbers, and this in turn forms an expectation and/or habit that we impose on physical forms that are not part of the body, like the tally, once we start using one. Nonetheless, it is possible to make regular parallel lines for a purpose like decoration, so the real challenge for archaeology lies in trying to discern the social purpose of linear marks whose regularity suggests intentionality but provides no clues as to social purpose, use, or meaning. Tallies do not have to be external to the body, though using the body like a tally does not provide advantages like morphological plasticity and the ability to accumulate and distribute cognitive effort that a material tally does. As we saw in earlier chapters, number systems in Papua New Guinea use the body as a tally: After finishing with the fingers on one hand, these systems typically move up the arm to the wrist, elbow, bicep, and shoulder; then they count up around the head, find a center point like the nose, and reverse the count down the other side of the body, ending with the fingers of the other hand.8 There are a few qualities worth mentioning for this type of counting. One is that these systems can be socially privileged, knowledge that is reserved for adult males and limited or proscribed

8

Wassmann & Dasen, 1994; Saxe, 2012, Owens, 2018.

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     for everyone else.9 It is easy to see why that might be: When the material device for counting is a human body, it cannot be passed around from person to person like a stick can be (this is distinct from collaborative forms of counting, in which participants use their individual bodies). It is also common to have social norms, typically prohibitions, regarding who is allowed to touch someone else’s body, particularly in public, as well as what kind of public touching is allowed. There are also norms regarding who can imitate someone else’s behavior, especially the behaviors of people with high social status. When the device in question consists of an adult male body and adult male behaviors, there is a good probability that other adult males might be the only ones allowed to witness and imitate the behavior, at least publicly.

      An interesting aspect of body-counting, one that reveals the tally’s influence on numerical organization and structure, is that the bodyparts tapped or otherwise indicated in counting (Fig. 4.1) and the verbal labels for the body-parts (Table 4.1) are both ordinal sequences. That is, they are elements with a specified order, they express numbers whose primary relation to each other is more-than, and the numbers lack the elaborated relations that characterize Western numbers (as for example, sixteen is seven more than nine). When given a Western-style subtraction problem to solve with their ordinal sequences, Oksapmin respondents created a complex doubleenumeration strategy: Subtracting seven from sixteen, 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 To exemplify the process, with sixteen minus one, “the thumb (1) goes with the ear-on-the-other-side (16), leaving 9

Wassmann & Dasen, 1994.

10

Saxe, 2012, p. 86.

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       the eye-on-the-other-side (15)” as the answer; for sixteen minus two, “the index finger (2) goes with the eye-on-the-other-side (15), leaving the nose (14)” as the answer.11 Oksapmin double-enumeration has important implications. One is that the need to develop such a strategy shows that numbers do not emerge preloaded with relations. That is, the Oksapmin numbers seven, nine, and sixteen are not related beyond their order in the counting sequence: Sixteen is not nine more than seven or seven more than nine, nor is it three more than thirteen or any of the other relations we might think of. Rather, it is the number between fifteen and seventeen and has no other relations. Nor can ordinal numbers be manipulated by means of those (absent) relations, since methods for calculating, physical or mental, have not developed; this is not surprising, since calculation needs more relations than an ordinal sequence provides. If the need to add and subtract were indigenously developed, rather than being imposed through cultural contact, the cumbersomeness of double-enumeration – for example, its costliness in terms of resources of attention and working memory – would motivate change toward forms and methods with greater efficiency, improved informational accessibility, and reduced demands on working memory and attention. These qualities would likely be achieved by incorporating a new material form, one selected because it shared the ability to accumulate while additionally providing the ability to manipulate. The material and linguistic forms used in the Oksapmin number system are also mutually reinforcing. Neither would be capable of generating relations or calculation methods on its own; a material form with manipulability would be required to explicate relations and manipulate numbers beyond the conceptualization of ordinally sequenced numbers as simply more-than the preceding number. As a result, without a motivating social need for calculating, the use of noncontrastive material and linguistic forms is likely to 11

Saxe, 2012, pp. 86–87.

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     yield long-term stability, especially for a number system associated with a small, relatively isolated group with little internal or external complexity to manage (this is also true of the Desana digit-tally system discussed below). And indeed, body-counting systems of Papua New Guinea may have persisted for a substantial period of time, given that they span multiple language families across an isolating geography.12 Moving off the body to a piece of wood or string not only makes numbers more public and more widely accessible within a social group, it also makes them more permanent and increases their capacity. Once a lot of notches (or knots) are accumulated on a stick (or string), new difficulties will emerge: Perhaps the number of notches will need to be altered in some way – rearranged, grouped, or removed – rather than just being accumulated. At this point, the lack of manipulability will be a problem. Or perhaps the number of notches will need to be verified. Since the ability to appreciate quantity just by looking is limited to about three or four, verification means counting the notches. Counting assumes that a counting sequence is available for this purpose, which might not be the case. Counting is also tedious, especially for elements that are minute in size or very large in quantity, as we will see when we discuss specific Palaeolithic artifacts. When a counting sequence is not available, the alternatives are either accepting the total as many or comparing the individual elements, one by one, to a known standard. The first is imprecise, the second laborious, and both would motivate finding another method of achieving the same goal, one that is more accurate and reliable and less costly.

    ()  The Desana are an Amazonian people who inhabit the Upper Rio Negro cultural area (Fig. 6.2). Their number system is restricted and counts to twenty using the hands and feet. This type of number 12

E.g., Lean, 1992; Owens, 2018.

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       system is known as a digit-tally system: The fingers and toes are the only material form used for counting, and these comprise a tally in which each finger or toe is analogous to a notch on a stick, with a key difference being that fingers and toes are used to express number, while tally notches are made to represent number. The associated verbal labels (phrases describing the hands and feet) and number concepts (sequences that accumulate ordinally with natural groups based on the hands and feet) are closely associated with the material form (the fingers and toes).13 Desana numbers are typical as digit-tally systems go (Table 4.3). The numbers one through three are unanalyzable, so they do not give any hint to the distributed exemplars they were once likely based upon; the term for four is a compound meaning two pairs. The terms for five and ten mean one hand and two hands, respectively; these were once likely the first nonsubitizable numbers. The numbers six through nine involve all the fingers on one hand and the requisite number of fingers on the second hand, which is a typical strategy for naming the gap numbers between five and ten. Beyond ten, the feet were recruited, a logical extension of counting to a similar material structure with a small technological distance. Terms for eleven through nineteen are named for both hands plus the appropriate toes, first on one foot and then on the other, as is typical in digit-tally systems. As is also typical for digit-tally systems, the term for twenty means feet finish, meaning in context that all twenty digits of a person have been counted. The quantifier maha, a lot, is used to convey the idea of values higher than twenty.14 The Desana are not known to count beyond twenty and do not use twenty as a productive value, so their numbers are classified as a quinary (base 5) system.15 The Desana do not appear to use material tallies, or at least if they do, the behavior is infrequent enough that the behavior has not been noted, not even in a 2006 indigenous account of a system of numerical signs that represents a recent, probably nineteenth-century, 13

Miller, 1999; Silva, 2012.

14

Miller, 1999, p. 46.

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15

Miller, 1999; Silva, 2012.

     adoption of Western decimal numbers and the idea of writing by the Desana.16 Interestingly, the way that the Desana numerical signs are constructed implies the absence of tallying, defined as the accumulation of identical elements placed in one-to-one correspondence with whatever is being enumerated.17 Instead of identical elements, the numerical signs represent cultural images and ideas. While elements might still be accumulated to represent numerical totals, each element differs morphologically from its fellows, creating unique combinations that must be memorized; the combinations can also differ in their quantity from the number they express, as for example, a three-pointed sign is used to express the idea of four men.18 Within the Upper Rio Negro (Fig. 6.2, upper right quadrant), peoples neighboring the Desana have been reported to use tallies. For example, the Baniwa (who speak an Arawakan language) and Yuruti (who, like the Desana, speak an East Tukanoan language) were described as incising flat pieces of wood along their edges to keep track of days in a journey (Fig. 10.1).19 Significantly, these devices were made in the context of work performed for a European employer, making cultural transfer highly probable. The absence of reports of tally use in the region suggests that the European practice did not catch on, a limited proliferation consistent with the prevalence of restricted number systems and the lack of social needs for greater capacity and more permanent recording. Mnemonic devices, including khipus, have also been described for the Upper Rio Negro: For example, the Warekena (who, like the Baniwa, speak an Arawakan language) have reportedly used them.20 None have been documented for the Desana, and Amazonian khipus are described as mnemonic devices used to recall (nonnumerical) information.21

16 17 18 19 20 21

Fernandes & Fernandes, 2006; also see Overmann et al., 2022. Overmann et al., 2022; also see Schlaudt, 2020. Fernandes & Fernandes, 2006; also see Overmann et al., 2022. Koch-Grünberg, 1921, p. 298. De Sampaio, 1825; Chaumeil, 2005; Hugh-Jones, 2016. Chaumeil, 2005; Hugh-Jones, 2016.

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      

. . Pieces of incised wood made by Baniwa and Yuruti men to keep track of work performed for a European employer in the nineteenth century. These devices most likely represented cultural transfer, especially given the general absence of indigenous tallying practices in the region. Adapted from Koch-Grünberg (1909, Image 61, p. 117).

Given their use of fingers and toes in tallying, it is interesting that the Desana do not use material (noncorporeal) forms for this purpose. As the restricted extent and relative youth of their number system imply, they do not appear to have a social need to record numbers (persistence) higher than what digit-tallying affords (capacity). That is, the social conditions motivating the recruitment of a material form with greater persistence and capacity than what the fingers provide are absent. But beyond this, the Desana do not seem to make linear marks for other social purposes. Linear marks can be mnemonic, conventional, decorative, utilitarian, or represent tools of various types.22 For whatever purpose they might be made, linear marks constitute a form that might easily be adapted to a numerical purpose, and which has the potential to inspire the idea of doing so. The lack of making linear marks for any purpose means that there is no easily adaptable material form or source of inspiration in the Desana cultural environment. This might tend to increase the technological distance between the emergent problem of needing greater persistence and/or capacity than the fingers/toes, and solving it by means of a material tally.

22

Reese, 2002.

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     Another factor may be that the fingers (and toes) are individuals. For example, the thumb is the biggest and opposing, the index finger is strong and used in pointing, the middle finger is usually the longest, and the ring finger tends to move with the little finger, which is the smallest and positioned at the edge of the hand. This individuality means that each finger or body part used for counting may not be considered to be “of the same sort” as the fingers or body parts that precede and follow it in a counting sequence.23 The associated numbers, which are conceptually and linguistically individuated by the fingers/toes used, are likely similarly individuated (Table 4.3). In contrast to the fingers and toes, the notches on a tally or the beads on a string can be deidentified or tokenized24 – where a token is a repeatable physical instance of a type and a type is a more generalized construct or class.25 Each notch or bead becomes a repeated instance of “not just ‘something’ but the ‘same thing’.”26 This difference between individuated forms like the fingers and tokenized forms like tallies and beads might explain why the toes, as well as the digits and bodies of other people, are so often used to extend counting beyond ten and twenty: While they too are individuated, they represent continuity in doing more of what is already being done, a relatively small technological distance. The use of individuated forms like the fingers might tend, at least initially, to inhibit the use of a material form like a tally for numbers, where each notch is made to represent a specific item. In this case, the tally would represent discontinuity in doing something new, a larger technological distance. Minimally, making notches for a nonnumerical purpose like decoration or music produces a material form with the potential to be noticed, recruited, and adapted to a numerical purpose. Further, decorative notches, even if not numerical in their intent, would likely be tokenized, as each would be another of the same thing. Whether they begin as individualized/numerical or

23 26

Schlaudt, 2020, p. 641. Schlaudt, 2020, p. 635.

24

Schlaudt, 2020.

25

Wetzel, 2018.

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       tokenized/nonnumerical forms, tallies instantiate visual and tactile regularities, since the elements resemble one another; ordinality, since each element has a relation to those preceding and succeeding it; cardinality, as the total number of elements; linearity, which is implicit in orderly arrangement; and succession, since each element is more than and potentially one more in quantity to the element it follows. The process and implications of tokenization are discussed further in the next section. The Desana case also shows that numbers can be available – perhaps for significant amounts of time, given the potential for longevity that emerges from using noncontrastive material and linguistic forms – and can become elaborated well beyond the emergence of the first subitizable numbers without incorporating the kinds of material forms that would leave some trace in the archaeological record. The things that the Desana currently use for their numbers – fingers and toes, gestures, and verbal descriptions – would be archaeologically invisible, as would the kinds of things that would typically follow them: the hands and feet of other people, body paint and marks on other surfaces, material forms made of organic (perishable) substances like wood and fiber, and unmodified material forms like pebbles or seeds collected for the purpose of counting but unrecognizable as such when not in use. This highlights one of the signal difficulties facing the archaeological search for numerical origins: We are unlikely to find an exact origin for numbers because the kinds of things used in emergence and initial elaboration are such that they would not leave any trace for us to find. Another archaeological challenge highlighted by the Desana case is that artifacts can be one-offs, isolated in time and space, like the Baniwa and Yuruti counting sticks and Warekena khipu. Uniqueness increases the difficulty of interpreting the function(s) that the artifacts would have represented; conversely, the likelihood of discerning function is improved when multiple similar artifacts are available to serve as sources of data and comparisons. This is especially true in notational systems because single artifacts are unlikely

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     to contain enough marks to identify and characterize a notational system, let alone decipher it. Notably, the absence of linear marks made for any purpose implies a greater technological distance between need (the ability to represent numbers with greater capacity and persistence than what the fingers can provide) and solution (a form that accumulates with greater capacity and persistence than the fingers have) because of the absence of sources of inspiration and easily adapted forms.

 Each finger or body part used for counting differs morphologically from the other fingers or body parts used in counting. Noting the importance of tokens in mathematics, where a repeated instance of “|” is semantically meaningful (| | |) in a way that a repeated instance of “the” is not (the the the), the philosopher Oliver Schlaudt has suggested that their physical differences individuate the fingers and body parts when used as counting elements.27 Essentially, they are not tokens. He contrasts this with elements that are repeated, visually indistinguishable, and imposed on a medium that does not constrain, limit, or influence the repetitions to be visually indistinguishable. These are tokens because they are not just a repeated instance of “something,” like the fingers presumably are, but a repeated instance of the “same thing.”28 For the purposes of this discussion, we will call the former type “individuated” and the latter “tokenized.” Schlaudt’s distinction implies that numbers based on digittallying would differ conceptually from numbers associated with tokenized elements because the fingers/toes are not tokenized. For the sake of discussion, we will reserve judgement on the merits of the propositions that the fingers/toes, at least in emerging number systems, are not understood as the same thing, and that morphologically distinctive tokens (a, a, A) cannot comprise a type (‘a’). It is worth noting that these do more than severely underestimate the abilities 27

Schlaudt, 2020, p. 641.

28

Schlaudt, 2020, p. 635.

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       for categorizing and abstracting: While the morphological distinctions make a difference in descriptive naming, they do not appear to make any conceptual difference in how the fingers/toes are used in counting. Under Schlaudt’s distinction, forms used as tallies or which are adaptable for tallying might be categorized as follows: Fingers and body parts would be classified as individuated because they are morphologically distinct, if not individuated by the specific item it represents. Notches, even if visually indistinguishable to the naked eye, would also be classified as individuated when each one is made to represent a specific item. In contrast, identical notches made for decorative or utilitarian purposes, while not numerical, would only be tokenized if their regularity emerged from a conscious intent to produce another of the same thing, rather than regularity being imposed by the medium.29 Since linear marks are the form most easily inscribed on the surfaces of sticks and stones, their regularity may represent constraint, not conscious intent. Given the prevalence of finger-counting in numerical emergence, this distinction raises the following questions. When a material tally is recruited into a system of numbers based on the fingers, like that of the Desana, are its notches necessarily as individuated as the fingers presumably are? It would make sense if they were, given the likelihood that the existing numerical concepts, which are based on the individuated fingers, would be extended to the new material form, at least initially. Under what condition(s), then, might the notches subsequently become tokenized? Would the condition(s) include repetition in their manufacture, frequency in their use for counting, their sheer prevalence as common cultural forms, the exactness imposed on their form, their lack of visual distinguishability to the naked eye, or the lack of any constraint that would force their original associations with specific items to be the only possible way of using them to recall those items? It seems reasonable to think that tokenization could emerge from the process of making notches, as each notch has the potential 29

Schlaudt, 2020, p. 635.

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     to be both “the next one” and “the next [one] of the same sort,”30 whether it is being made for a numerical or nonnumerical purpose and regardless of whether or not or the degree to which the medium it is being inscribed upon might constrain it. Prehistoric notches, it should be noted, can be so morphologically regular that any differences can only be distinguished under the microscope.31 The subsequent use of conceptually individuated notches – away from their originating circumstances in terms of time, distance, and the intentions, understandings, and memories of the people who created them – might also tend to decontextualize or reify their individual meanings, causing them to tokenize. Certainly, nothing in the actual form of linear marks identifies or even hints at what they originally counted, and nothing in their visual indistinguishability would influence their conceptual individuality toward continuity and persistence. Their deidentification as individuals and acquired meaning as tokens, in other words, would emerge through their processes of manufacture and use, and through being the material forms that they are. Schlaudt proposes a different mechanism, seeing the conditions enabling tokenization as involving repetition and exactness, but only when these qualities “stem from a conscious intention” to produce them, rather than being “imposed by the material constraints of the symbolic technique.”32 While we might think that consciously imposed repetition and exactness would surely include linear marks made for both recalling specific items and decoration, linear marks are arguably constrained by the manufacturing technique as the form most easily imposed on wooden and bone surfaces.33 Against the view that mark-making is the kind of material engagement through which such meanings can and do emerge,34 this restriction seems to eliminate making linear marks for any purpose as a mechanism for tokenization. With linear marks disqualified, the mechanism for tokenization that remains apparently occurs through artistic forms with no obvious

30 32

Schlaudt, 2020, p. 641. Schlaudt, 2020, p. 635.

31 33

D’Errico, 1991; d’Errico & Cacho, 1994. 34 Schlaudt, 2020, p. 635. Malafouris, 2021.

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       connection to numbers. Schlaudt finds two instances in the prehistoric record that fit his definition. Both are decorative, and neither is overtly numerical. One consists of “engraved depictions of animals on portable objects,” exemplified by a horse motif engraved on bones from the Madeleine Rock Shelter, France; the other involves “geometrical patterns in which the ‘fit’ of the strokes requires exact repetition,” exemplified by the geometric crosshatch design engraved on pieces of ochre from Blombos Cave, South Africa.35 While this definition widens the scope of material forms that might occasion tokenization, it also raises the questions of how and why tokenization realized through complex depictions or geometric patterns might transfer to devices made and used for numerical purposes, which cross-culturally have much simpler forms, like linear marks. Complex forms also seem like they would be more difficult to tokenize, relative to linear marks, especially if repetition and exactness are key criteria: Linear marks are easier to repeat and to repeat exactly than are complex forms like horses (crosshatches, being linear marks made in diagonally opposed series, are presumably intermediate in difficulty). At a more basic level, it is not certain that repeatability and exactitude in a horse motif or geometric pattern represents the same kind of tokenization found in numbers, where the addition of another token is meaningful because each repetition produces a different number and these comprise a series: |, | |, | | |, | | | |. . .. While a repeated horse motif might be said to comprise an analogous series, the labor needed to produce even a single form makes the complex motif inherently less repeatable and accumulable than a notch, and thus less likely to engender tokenization, if not concepts of number, relative to notches. As for geometric patterns, the relations are spatial and aesthetic, not numerical. In principle, it seems possible that tokenization, as a mental quality that “occurs in prehistoric art,”36 might subsequently be available to inform how numbers are conceived. Not only is this the 35

Schlaudt, 2020, pp. 637–638.

36

Schlaudt, 2020, p. 641.

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     kind of neurocentric construct we are trying to avoid, it is also challenged by contemporary ethnographic observations: Societies like the Desana produce complex forms of art that surely occasion tokenization as outlined by Schlaudt. These include petroglyphs and cultural motifs later adopted into an indigenous system of writing, where the forms are exactly repeated in media that impose no constraints that might influence exactness.37 However, Desana numbers are fingerindividuated and thus, if Schlaudt is correct, not tokenized. This shows that numbers can develop without tokenization, even when it is present in domains like art, and it also suggests that tokenization realized through art does not automatically transfer to numbers. More broadly, evidence of tokenization in prehistoric art does not entail that any numbers present were necessarily tokenized, nor indeed that numbers were even present. This has direct relevance to interpreting linear marks on Palaeolithic artifacts, as the presence of tokenized art does not mean that contemporary linear marks were numbers, tokenized or not. In trying to infer prehistoric numeracy from nonnumerical artifacts, Schlaudt has created a mental route to numbers – art first, then tokenization, and then the ability to produce marks that are tokenized and numerically meaningful – that differs substantially from the process of emergence observed today, in which the perceptual experience of quantity is expressed and manipulated by means of distributed exemplars, the fingers, and devices like tallies. Further, the production of marks for any purpose creates motor and visual regularities with the potential to occasion tokenization. If this is correct, then tokenization either will not apply to prehistoric or contemporary emerging number systems, no matter how culturally prevalent their tokenized forms of art might be, or it will be possible for tokenization to emerge through processes like making marks for individuated series or repurposing nonnumerical marks, forms and mechanisms

37

Alemán et al., 2000; Fernandes & Fernandes, 2006; Hugh-Jones, 2016; Overmann et al., 2022.

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       with advantageous qualities like simplicity and repeatability and which correspond to contemporary observations of the kinds of forms used in numbers. In either case, we end up in the same place we were before the individuated/tokenized distinction was made: We currently lack a way to discern the meaning of undifferentiated linear marks, unless either someone from the culture that made them tells us what they are supposed to mean or we have well-documented analogies from contemporary cultures to which they might be compared. The archaeological record is unlikely to show us much, if anything, about prehistoric numerical emergence because of the use of perishable materials, and prehistoric linear marks cannot conclusively be identified as numerical. Tokenization tries to work around this problem by inferring numerical thinking from nonnumerical forms. Such attempts highlight another benefit to cultural analogy as discussed in Chapter 6: empirical verifiability. That is, the idea that artistic forms can reveal tokenization that then informs numbers and numeracy can be empirically tested by looking for similar associations in contemporary cultures. Until this has been done, contemporary societies with tokenized art but individuated numbers suggest that art tokenization and numerical tokenization are severable phenomena, which implies that any tokenization found in prehistoric art does not and cannot reliably signal ancient numeracy.

   In the absence of number concepts or number-words, people use oneto-one correspondence, a tallying behavior that associates the members of an enumerated set with those of a reference set, such that all of the members of both sets are matched one to one and none are matched more than once. The modern rosary is an example of such tallying, as using the device associates each bead with a prayer in a way that does not involve number-words. Another example of tallying in the absence of explicit concepts or words comes from the memoir of William Buckley (1780–1856), a transported convict who escaped to

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     live with the Wathaurung people of southern Australia for over thirty years. While some elements of his story are fantastical and were perhaps calculated to raise interest in his published account, both contemporary and current scholars have noted that Buckley’s account accurately described aboriginal life.38

[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. (Morgan, 1852, pp. 48–49)

Just over fifty years later, the Wathaurung number system was still being described as restricted in its extent, with terms recorded for one and two, ku-i-muty and bullaity.39 This estimation is likely accurate because a 1998 source added only terms for three and several, kulik and getjawil:40 The restricted extent for Wathaurung numbers is also consistent with the restricted extent of Australian number systems generally, as a 2012 sample of 189 Australian languages representing thirteen language families and three linguistic isolates showed that most counted no higher than three or four.41 Tallying without words, like using rosary beads to keep track of prayers or stripes of mud to keep track of the days of a journey, is found in both societies with highly elaborated numbers and ones with few numbers. In the Wathaurung case, the tally was used in a society whose language had no number-words numbers higher than two or three. This suggests something different than what we saw with the Desana, where the transition from individuated digit-tallying to tokenized material-tallying involves not just technological distance but an absence of adaptable forms to act as prompts. With the Wathaurung, the questions are different: Are elements like stripes of mud already

38 39 41

Curr, 1886b, pp. 57–58; Tipping, 1966; Flannery, 2017. 40 Mathews, 1904, p. 732. Blake et al., 1998, p. 71. Bowern & Zentz, 2012, pp. 133–134.

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       tokens, and if so, does this status then inhibit their later individuation by means of verbal labeling? From a different perspective, does the lack of verbal individuation and associated inability to count the elements entail that they are tokens? They might not meet Schlaudt’s definition, but on the other hand, it is difficult to imagine them as individuated: Each stripe does not mean that a particular day has passed or must later be recalled, but that another day has passed and that the return journey will take as many days. Where body-counting and digit-tallying involved noncontrastive, mutually reinforcing material and linguistic forms, the Wathaurung case may be a system in which numbers above the subitizing range comprise a material-only system that does not contrast because the material form, the ad hoc tally with tokenized elements, is unaccompanied by a linguistic form. To differentiate this from the system incorporating noncontrastive material and linguistic forms, as in the Oksapmin body-counting and Desana digit-tallying systems, we will call the Wathaurung material-only variant “uncontrastive.” Absent any emergent social need for the number system to do something other than what it is already doing, the uncontrastive system might similarly comprise a highly stable system of numbers. Longevity does seem to be the case when the available data on aboriginal number systems are considered.42 Long-term stability might also be influenced by the greater technological distance involved in conceptually and linguistically individuating tokens like stripes of mud, especially in the absence of any social need for such individuation. In essence, why have labels for the numbers associated with things like stripes of mud when they are so perfectly usable without them? The use of material forms may differ as well. In the 2012 sample of Australian languages mentioned earlier, some of the languages had terms for five, a number whose name, if analyzable, provides insight into the material form used to realize it. In this sample, sixteen 42

Bowern & Zentz, 2012; Zhou & Bowern, 2015.

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     languages had unanalyzable terms and are thus uninformative, twenty-three had analyzable terms based on the hand, eleven had analyzable terms that were compounds of subitizable elements (two plus two plus one or two plus three), and four used both analyzable types.43 These numbers suggest that, in comparison to the bodycounting and digit-tallying systems described above, some Australian number systems have not been elaborated to use the body and its individuated elements as a material substrate for counting to numbers higher than the subitizing range, but instead, may have adopted the strategy of leveraging tokenized environmental objects like notches or pebbles or stripes of mud. This highlights the effects of using particular material forms for counting on numerical outcomes. It also raises the possibilities that vocabulary-based methods might significantly underestimate Australian numeracy and that Australian number systems may show us what human numeracy is like when it is elaborated materially without involving language, at least above the subitizing range. Since material forms are not typically included in descriptions and analyses of cultural number systems, few studies currently consider how the material forms used in a cultural system of numbers might influence numerical content, organization, and structure, let alone how they might interact with verbal labeling in the historical development of a cultural number system, or how various combinations of material and linguistic factors might affect matters like elaboration and longevity. This lacuna is significant and unfortunately pressing because of the ongoing loss of number systems associated with minority cultures around the world, particularly through their displacement by Western numbers. If this interpretation of Australian numbers as tokenized and unnamed above the subitizing range is correct, the implications are interesting: At least some of what we see prehistorically could be the same type of system, rather than the more familiar type in which 43

Bowern & Zentz, 2012, p. 138.

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       numbers are individuated by means of the fingers, distributed exemplars, tallies, and verbal labels. Emergence might also consist of an initial phase involving the perceptual experience of quantity and the use of distributed exemplars, followed by a choice either to realize individuated nonsubitizable numbers or just use them in a tokenized manner. Several interesting questions are also raised: If not a choice between the two types, should we assume that tokenized systems are prior, and if so, how and why would the more effortful process of individuating numbers emerge from them? Are there really two types, or just one that has been described somewhat differently for Australia and South America? In addition, how far back in time might tokenized numbers have emerged? Tool-use goes back 3.3 million years that we know of,44 and this suggests that tokenized numbers, in not involving language, might have had the potential to emerge before language itself did. And what is the long-term prospect for the elaboration of tokenized numbers without names, given the perspective that language is essential to numbers? Finally, assuming that we can distinguish individuated and tokenized types of tallying by observing and describing contemporary cultural practices, how might we distinguish these archaeologically? Given the questions posed here and in the previous section, do we even need to do this? The question of when prehistoric societies began to use devices like tallies is already difficult for archaeology to answer, as there are no conclusive techniques or criteria for determining the social purpose, use, or meaning of marks on the few artifacts made of nonperishable substances, or whether noninscribed forms like beads were used for counting. Beyond the establishment of any use for counting is the question of what type of number system an artifact might have represented, a matter for which there currently are no diagnostic techniques. The archaeological techniques currently in use and their limitations are addressed in the next chapter.

44

McPherron et al., 2010.

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 Interpreting Prehistoric Artifacts

One of the singular challenges of prehistoric archaeology is interpreting the intent, purpose, and meaning of marks whose regularity of length, spacing, and orientation makes them visually indistinguishable from one another, like those in Fig. 11.1. Were they decorative? conventional? mnemonic? symbolic? notational? numerical? astronomical? calendrical? musical? utilitarian? And assuming they were numerical, what kind of number system might they represent? Would they represent one-to-one correspondence with no associated vocabulary, like the Wathaurung mud-stripe tallying? Or an ordinal sequence that corresponds to the terms in the language, like Oksapmin body-counting? Or would they represent specific objects, as a material tally recruited into the Desana digit-tally system might someday do? Prehistoric numbers most likely differed from ours, in being less elaborated, if nothing else. Given the widespread use of perishable materials like wood and fiber for numerical recording and calculating in contemporary societies, the earliest prehistoric numbers are likely to be archaeologically invisible. How elaborated might numbers have been by the time Palaeolithic societies begin to use materials like bone and stone to record them? Would these numbers have been fairly well codified and elaborated by the time more permanent materials were used, and if so, why are notations that are unambiguously numerical not found until much later, well after the Neolithic? There are a lot of possibilities, and currently no methods or criteria for deciding among them with the desired level of reliability and confidence. Nor is this the only shortfall. Another material form



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   

. . Engraved baboon fibula from Border Cave, South Africa, dated to 42,000 years ago (see d’Errico et al., 2018). The artifact is 7.6 centimeters (3.0 inches) in length, which makes the incisions approximately 0.15–0.25 centimeters (0.06–0.10 inches) in length. Adapted from d’Errico et al. (2012, Supporting Information, Fig. 9, top image). Image © PNAS and used with permission.

with the potential to be used for counting is the string of beads,1 as we saw with the Pomo.2 Beads have a much longer prehistory than the 30,000-year-old cache found at Sunghir might suggest.3 An example dated to 77,000 years ago is shown in Fig. 1.2, and even older examples are known: Beads from Skhul Cave, Mt. Carmel, Israel have been dated to 100,000–135,000 years ago,4 and beads from Bizmoune Cave, Morocco have been dated to 142,000 years ago.5 Beads that were strung and worn as ornaments have physical characteristics like punched holes, wear-marks left by string, and traces of ochre used as body paint that identify them.6 Nevertheless, these physical characteristics cannot indicate whether beads were also used as a rosary,7 let alone whether they were used in conjunction with number concepts and associated vocabulary. Many cultures make and wear beads, but few count with them, so prehistoric beads are more likely to have been used as personal ornaments.8 However, this cultural observation does not identify possible exceptions. For example, Sunghir stands out because of the number of beads, not because anything found at the site suggests they were used in counting. The emphasis on ornamentation also tends to keep prehistoric beads from being considered as

1

2 3 4 7

Coolidge & Wynn, 2011; they credit one of their students for bringing the idea to their attention. Kroeber, 1925; Loeb, 1926; Closs, 1993. White, 1999; Marom et al., 2012; Nalawade-Chavan et al., 2014. 5 6 Vanhaeren et al., 2006. Sehasseh et al., 2021. D’Errico et al., 2005. 8 D’Errico, 1998. Overmann, 2013b.

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     potential counting devices in the first place. On similar grounds, hand stencils like those at Cosquer are also generally excluded from consideration. Currently, the only artifacts routinely being analyzed for possible numerical use are pieces of bone (or similarly hard and inscribable materials like ivory and stone) containing marks. When those marks are regularized incisions, the artifact is often assumed to be a tally, and the marks are assumed to represent numbers that are mentally realized and which are akin to Western numbers. If the number of marks is around twenty-seven to thirty, the tally is usually assumed to represent the days in a lunar or menstrual cycle.9 Resemblance to the modern tally, whether physical or numerical, is a risky criterion,10 as we will see with the Australian messenger sticks. And even if a particular artifact has correctly been identified as a numerical tally or lunar calendar, it cannot tell us anything about the kind of number system that produced it, since the cultural analogies that we might draw upon for this purpose are so poorly documented.

As

noted

previously,

notational

devices

are rarely

photographed, illustrated, or explained in terms of what they meant or how and why they were organized.11 Further, the insight we do have is rarely correlated with the associated number systems, so we do not know whether and the degree to which marks that mean numbers might correspond to numerical properties like organization and structure. Presumably, marks associated with highly elaborated numbers would differ in characteristic ways from those associated with emerging numbers. To date, no such analysis has been performed, and the data required might not even be available in the quantities and details needed. In this chapter, we will look at what is involved in determining the social intent, purpose, and meaning of regularized but

9

10

Marshack, 1972, 1989, 1991a, 1991b; Beaumont, 1973; Thompson, 1981; d’Errico, 1989, 2001; Zaslavsky, 1992; d’Errico et al., 2012. 11 Malafouris, 2021. Hayden, 2021, p. 2.

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    undifferentiated marks inscribed on the surface of a bone. For this, we will review some of the archaeological and other techniques used, as well as their assumptions and the challenges of applying them to Palaeolithic artifacts. We will look at a contemporary model, the messenger sticks used by Australian societies, whose regular linearized marks, as interpreted by culturally knowledgeable informants, often have conventional, nonnumerical meanings. We will also look at some functional considerations: For example, what are the implications for any use in numbers when marks are very small in their size or very large in their quantity?

-    Mark-making first appears in the archaeological record as the inadvertent byproduct of processes like scavenging or butchery,12 behaviors practiced several million years ago by hominins – the category of bipedal apes that includes humans, species ancestral to humans like Homo habilis and the australopithecines, and species closely related to humans like Neandertals and Denisovans. The ability to make marks unintentionally was presumably followed, at some point, by the ability to make marks intentionally, but without any numerical meaning or purpose; this, in turn, was presumably followed, at some point, by the ability to make numerical marks. Unintentional marks would have provided the kind of manuovisual engagement and visual stimulus needed to enable their being made purposefully, since even inadvertent mark-making produces form that then becomes available for being noticed and responded to.13 Further, the manufacture and presence of intentional marks – again, made for any purpose – would have provided the kind of manuovisual regularities important in numerical patterning, and this would have increased the likelihood of marks being made for numerical purposes.14

12

13

De Heinzelin et al., 1999; Semaw et al., 2003; Domínguez-Rodrigo et al., 2005; McPherron et al., 2010; Harmand et al., 2015. 14 Malafouris, 2013, 2021. Overmann, 2016d.

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     Intentionally made marks encompass a broad spectrum of social purposes, intents, and meanings: private, utilitarian, aesthetic, and symbolic. Private marks are meaningful only to the individual making them, utilitarian marks are made to fulfill a function, and aesthetic marks are made for decorative or artistic purposes. Symbolic marks, in contrast, are made to stand for ideas, relationships, or objects.15 But as we will see with the Australian message sticks, symbolic marks too can involve a broad spectrum of meanings: a social purpose like ensuring safe passage for the messenger who carries them; a convention that shows affiliation with a particular social group; a mnemonic use to recall a narrative message or list of specific items; a calendrical use to record days or lunations; or a numerical use to quantify an amount of items. The challenge is deciding among these plausible alternatives because – for both prehistoric and contemporary numeracy – linear marks need not be intentional; if intentional, they need not be meaningful; and if meaningful, they need not mean numbers. The question is not so much when linear marks become intentional, then meaningful, and then meaningful as numbers, but rather, how these conditions might be discerned and differentiated in the archaeological record. The species making the earliest marks were not human, so contemporary notational systems are an inexact model at best. Here we might recognize mark-making as more than a sign of minds becoming human, since it is the kind of material engagement through which such change occurred.16 Comparisons involving artifacts dated to the last 300,000 years, the period when fossil skulls begin to resemble those of modern humans in their shape and size, are on firmer ground in this regard; even more so are artifacts dated to the past 30,000 years, given unambiguous signs of socio-material conditions that would motivate numerical elaboration. The shortfall then becomes the absence of detailed cultural analogies and a solid understanding of

15 16

Deacon, 2012; also see discussion in Coolidge, Overmann, et al., 2023. Malafouris, 2021.

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    how and why contemporary societies use material forms as numerical tallies and how those tallies represent numbers and reflect actual number systems. Based on close examinations, descriptions, and analyses of prehistoric bones with inscribed marks, the archaeologist Francesco d’Errico and his colleagues have sketched out a five-stage scenario: (1) producing marks; (2) producing identical marks; (3) attributing meaning to individual identical marks; (4) adding similar marks at different times; and (5) using differences in the morphology, spatial distribution, accumulation over time, and quantity of marks to comprise a code.17 This scenario spans the temporal range from the earliest prehuman marks to those made by Upper Palaeolithic societies. It does not consider how number systems are observed to emerge today (i.e., through ephemeral means like fingers), nor the fact that some are still in the process of emerging. The thorny question these omissions raise is this: Just how conflated are the prehistoric emergence of numbers and the evolutionary development of the human mind? We want a single universal process that explains numerical emergence and elaboration in both prehistoric and contemporary societies; at the same time, we want to separate the deep evolutionary history from the universal process because otherwise, we end up with separate processes for prehistoric African/Eurasian numbers and contemporary emerging ones. We should assume that prehistoric societies used distributed exemplars and finger-counting prior to using material forms, and that they used material forms made of perishable substances before they used ones made of preservable materials. Not only are these what contemporary societies do, it is more parsimonious to think that prehistoric and contemporary numbers emerged through the same processes than through different ones. While we do not have evidence of prehistoric material forms made of perishable substances being used for numbers, we do know that the use of perishable substances 17

D’Errico et al., 2018.

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     for cultural purposes is a significant lacuna in what the archaeological record is able to reveal. We also have some indications of prehistoric finger-counting, as proto-language analyses show the hand being used, both as etymologically transparent forms that reference the hand and in organization by amounts based on the hand. While this evidence occurs relatively late in the Palaeolithic, it does suggest that ancient numbers emerged in the same way we see in contemporary number systems. Disentangling the prehistoric emergence of numbers from the evolutionary development of mark-making is possible. We need to consider separately the mark-making and other material culture that produced the human mind over the last two or three million years from the much more recent emergence and elaboration of cultural number systems. Yes, the two are related: The ability to make marks and imbue them with meaning is the evolutionary heritage of all human societies. We not only find artifacts that reflect this evolutionary history, they are critical to understanding the development of the abilities to make marks for any purpose, including numbers, and leverage material forms for cognitive purposes.18 However, the nexus of psychological abilities, behavioral capacities, and socio-material conditions necessary and sufficient for the emergence and elaboration of numbers did not occur until relatively late, perhaps some 50,000– 30,000 years ago, with the development of the necessary socio-material conditions being postponed for the populations migrating into the remotest continents. This admits the possibility of a single process whereby contemporary and prehistoric numbers emerge and elaborate through the use of ephemeral forms like fingers and gesture and the later recruitment of material forms like tallies. Mark-making becomes part of an extended cognitive system for numbers, not the sole witness that attests to a prehistoric mind capable of conceiving them. As it stands now, with the prehistoric emergence of numbers and the evolutionary development of mark-making conflated, there is 18

Overmann, 2021f.

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    a tendency to claim that earlier and earlier marks represented numbers; an example is the recent claim that Neandertals had numbers in the Middle Palaeolithic.19 Even if prehistoric numbers and evolutionary mark-making are separated, there are currently no reliable techniques for differentiating them archaeologically because a series of undifferentiated linear marks carries no information about its social purpose, intent, or meaning. And as previously noted, we currently lack detailed cultural analogies that might provide insight into how numerical marks might reflect a society’s number system, and when and why a society might incorporate a durable substance like stone to record its numbers. Prehistoric artifacts with marks imply, minimally, the availability of material forms (e.g., made for decorative, utilitarian, or conventional purposes) suitable for being recruited and adapted for numbers. Nevertheless, the society using such forms might not have had the need to represent their numbers with any capacity or persistence greater than what their fingers would have provided. Under these conditions, an emerging system of numbers would most likely remain restricted – to count no higher than twenty or to smaller amounts like three or five, as is the case today for many of the number systems of Australia and South America. As the Australian message sticks will show, societies with restricted number systems can have a rich tradition of making artifacts whose notations convey a range of nonnumerical meanings. Alternatively, an elaborated system of numbers based on relatively perishable organic forms like sticks and knots, like those of the Pomo, might not record their numbers on a more durable substance like stone. Thus, they would leave no archaeological trace behind, despite the elaboration of their numbers. While we do not currently know why any society chooses bone or stone for recording its numbers, the notations might distinguish an elaborated system of numbers from an emerging one. Only the former, we might predict, would include unambiguously numerical 19

D’Errico et al., 2018; Barras, 2021.

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     characteristics like grouping that are nascent or absent in the latter. Since unambiguously numerical notations do not emerge until very late (the clay impressions of mid-fourth millennium BCE Mesopotamia; see Fig. 7.1), several possibilities are suggested. Perhaps there were no or few numbers; this seems unlikely because cases like Sunghir have the socio-material conditions likely to motivate the use and elaboration of numbers. Or perhaps there were elaborated number systems, but all of them used only perishable materials for numbers. This seems more likely, not just given the requisite socio-material conditions but contemporary cases like the Pomo who recorded their elaborated numbers with wood and fiber. But since the motivating conditions precede any elaboration that develops in response, we also cannot exclude the possibility of large and connected societies with relatively unelaborated number systems. We should also ask what kind of notations an elaborated number system would produce. The Pomo, for example, if they had recorded their numbers on bone, would most likely have made marks that resembled the sticks used in counting: one small line (the small stick worth four strings of beads), one large line (the large stick worth five small sticks), and one large line differentiated in some way (the large marked stick worth ten large sticks). These hypothetical notations would be unambiguously numerical because of the inherent relations: They would have unbundled maximums, the highest quantity that lower values reach before occasioning replacement by the next higher value. For example, as five small sticks would occasion replacement by one large stick, the notations should never contain more than four marks representing the small sticks. Reconstructing such numerical relations ideally requires multiple artifacts, as a single artifact might not contain, for example, nine as the unbundled maximum of the mark for large sticks. Since many Palaeolithic artifacts are one-offs, which hampers this kind of reconstruction, a one-off of these hypothetical artifacts would contain up to three types of lines. However many were present, these would likely be consolidated (like with like), and organized by magnitude (largest to one side, smallest to

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    the other, intermediate in between), just as the sticks themselves were, because this organization makes the numerical information more accessible, relative to random order. However, no Palaeolithic artifacts have been described as using such encoding. This suggests either that none of the artifacts found to date reflect elaborated numbers, or perhaps that notational conciseness can develop subsequently to the use of marks. This would likely be governed by the material forms used, the frequency of use, and degree of numerical elaboration. While the hypothetical Pomo notations would most likely resemble the counting sticks (and thus reflect their conciseness, as achieved through bundling), the possibility that notational conciseness might have developed subsequently is mentioned in conjunction with the Ishango artifact below. Let us assume, briefly and for the sake of discussion, that prehistoric marks on nonperishable materials like bones were numbers. These numbers most likely would have been realized and elaborated through the use of the fingers (and toes), as the hands/feet are the material form most commonly used today in emerging number systems. These numbers would also likely have been in existence for some period of time – long enough to motivate not only their being recorded, but also the use of a nonperishable substance for it. Given these conditions, some characteristics indicating numerical elaboration should be apparent in the marks themselves. For example, the perceptual limits on appreciable quantity would at some point influence the use of grouping for number systems that frequently involved the need to make more than three or four marks. But with one or two ambiguous and highly debated exceptions – the Ishango and Les Pradelles artifacts, discussed below – no prehistoric artifact has marks organized in ways that unambiguously identify them as numbers. In fact, as previously mentioned, the earliest notations with unambiguous numerical organization, the Mesopotamian impressions, occur very late, only 5000 or 6000 years ago. The late emergence of unambiguously numerical marks suggests that Palaeolithic marks either did not represent numbers (which would not exclude the

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     possibility that societies had numbers, as in Desana digit-tallying) or represented unelaborated numbers (perhaps the kind of tallying represented by the Wathaurung mud stripes). In interpreting prehistoric marks, it is important to keep in mind that marks made for numerical purposes would not necessarily represent numbers that were identical or even highly similar to ours, given the degree of elaboration true of today’s Western numbers. Unfortunately, the assumptions that linear marks have numerical meaning20 and that numerical meaning entails numbers like ours are deeply entrenched, and as a result, archaeological analyses have not considered possibilities like the Desana and Wathaurung cases or the implications of cultural analogies generally for prehistoric numeracy.

      Using methods pioneered by Francesco d’Errico, determining whether marks were made intentionally (or not) is a relatively straightforward matter: Marks made unintentionally are more likely to be haphazard and thus irregular in morphological characteristics like length, orientation, and disposition across the artifactual surface, while intentional marks are much more likely to be regularized in these characteristics. Regularity and irregularity form a spectrum in which artifacts that are more extreme in their characteristics are relatively easy to designate as intentional or unintentional, while artifacts more intermediate between the two extremes are correspondingly more difficult to classify. To some degree, discerning haphazardness or regularity is a judgment call, perhaps as much art as science. Judgments are also insensitive to degrees of skill: That is, marks made intentionally but haphazardly because of a lack of proficiency are less likely to be identified as intentional.

20

D’Errico et al., 2018, p. 7.

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    For marks that are deemed intentional, determining the social meaning, purpose, and use is a much less straightforward matter, particularly when the marks are undifferentiated. Conceivably, undifferentiated or regularized marks could fall into several different categories: unintentional marks that just happen to be fairly regular; intentional marks made for nonnumerical purposes; intentional marks made for one-to-one correspondence without explicit numerical concepts; intentional marks made to recall a series of items; and intentional marks with an explicit numerical meaning within an elaborated system of numbers. Determining whether intentional marks were notations involves comparisons with experimental and ethnographic data, again using techniques pioneered by d’Errico. While experimental data are produced by Western researchers, the ethnographic data are described as consisting of notational systems from Africa, Europe, and the Americas. By analyzing these notational systems, d’Errico has identified four encoding factors used to differentiate notational marks from one another: The encoding factors are temporal accumulation, which looks at whether marks were made at a time or accumulated over time; quantity, which describes how many marks an artifact contains; morphology, which describes the form of the marks; and spatial distribution, which describes where the marks are relative to each other and the artifactual surface.21 Discerning

temporal

accumulation

involves

determining

whether marks were made in a single session (at-a-time) or multiple sessions (over-time). At-a-time accumulation is assumed more likely to represent a nonnumerical purpose like decoration, over-time accumulation to represent a numerical intent. The temporal distribution of mark-making is inferred from incisional characteristics indicating the use of the same or different tools, similar in concept to ballistics testing, where the striations left on a bullet can be matched to the gun barrel that produced them. In the case of prehistoric marks, the 21

D’Errico, 1998.

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     striations left by a tool are unique and thus identify the tool, and comparison of these characteristics can determine whether a single tool was used or several. The data collected from microscopic examination are then compared with experimental marks, which are made with the same or different tools under controlled conditions and thus provide an excellent and reliable standard.22 Nonetheless, there are several problems with the technique. First, it is not clear that the use of different tools and/or over-time accumulation entails that the marks were notational or numerical. That is, it seems possible to use the same tool to accumulate marks whose intent was enumeration, and different tools to make marks for decorative or other nonnumerical purposes. As the archaeologist Marion Prévost and her colleague recently noted, the “production of engravings either made in a single or multiple sessions is not an observation that can be considered in favor or not [for] a symbolic behavior.”23 Along similar lines, the archaeologist Brian Hayden has noted,

. . . accretion or periodicity (sensu marks accreted at regular intervals) is not necessarily a characteristic of all notations since single loan or debt tallies presumably recorded all items at once, while gambling or gaming scores could easily be created in one sitting, and notches representing the days until a certain event must have been made all at once. In addition, elaborate art or decoration is rarely consummated at one sitting, but is often completed over a series of days or weeks or longer periods, thus, exhibiting periodicity as well. (Hayden, 2021, p. 3)

Second and critically, when the resultant marks are indistinguishable except under the microscope, the premise itself becomes questionable. That is, prioritizing the regularity of the marks, yet

22 23

D’Errico & Cacho, 1994; d’Errico, 1998, 2001; Parkington et al., 2005. Prévost & Zaidner, 2022, p. 147.

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    distributing their accumulation across time and tools, might argue against the marks being numerical, since regularity to the point of visual indistinguishability is a greater investment in form than is required in accumulating marks for a numerical purpose, something known as overdetermination. In fact, spending the extra time to make highly regular marks would often be counterproductive in counting because it would encumber and slow the process with a particularly laborious form of recording. Cultural analogies would be helpful in this regard, since the degree of regularity imposed by contemporary peoples when making enumerative marks, or a preference for using the same or different tools when making marks for different purposes, seem empirically tractable. Discerning quantity, morphology, and spatial distribution starts with detailed descriptions of prehistoric marks, followed by their comparison with the ethnographic data compiled by d’Errico.24 Multiple issues perplex the technique. First and foremost, when the linear marks are visually indistinguishable, they do not comprise any kind of code, let alone a numerical code, so there is no assurance that their intention, purpose, or meaning was notational, let alone numerical. Second, it is not clear that d’Errico’s ethnographic data include non-notational marks made for social purposes like music, tool utility, or wayfinding, and this, in turn, implies that any prehistoric marks compared to them have been prejudged as notational. The omission of non-notational marks would also degrade the ability to use the ethnographic data to differentiate prehistoric notational and non-notational marks. D’Errico has associated the four encoding factors with six social purposes: prayer aids, memory aids, records, calendars, messages, and hunting marks.25 He has not correlated the encoding factors and social purposes in ways that might permit the latter to be inferred from the former. Further, he has not specified the encoding factors in terms of any numerical organization or structure, possibly indicating 24

D’Errico, 1998.

25

D’Errico, 1991, 1995, 1998, 2001; d’Errico & Cacho, 1994.

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     that there were none in the ethnographic data. A difference in the length or orientation of every fifth or tenth mark, for example, would suggest a numerical system, and it would be interesting to know how prevalent such encoding might be in contemporary notation systems. Finally, d’Errico has not specified the ethnographic data in terms of the associated number systems,26 and things like the extent of counting and organizing base are potentially useful to making inferences about a number system from the associated marks. An absence of numerical encoding, for example, would be understood differently if it were associated with restricted numbers than it would with highly elaborated ones. Admittedly, if the number systems in d’Errico’s ethnographic comparison were described, the data still might not be able to indicate anything about the number systems associated with marks, contemporary or prehistoric. This is because regularized linear marks are associated with both highly elaborated numbers, like those of the Western tradition, and relatively unelaborated numbers, like those found in Australia.27 The fundamental problem is this: Marks that are visually indistinguishable do not comprise a code of any kind. Thus, they provide no clue to their meaning, and they have the potential to encompass a wide range of meanings. They are not, on their own, informative about whether they were meant to be numbers, let alone the degree to which the society’s numbers might have been elaborated or whether the society even had numbers. Complicating the picture further are the facts that linear marks can be made both before and after numerical tallying emerges, and for both numerical and nonnumerical purposes after tallying emerges, and that tallying can coexist with a wide range of numerical elaboration. Significant socio-material complexity does not necessarily tip an interpretation one way or another because, as previously noted, both the making of linear marks and increased socio-material complexity are found in societies with few numbers and in ones with highly elaborated numbers. 26

D’Errico, 1991, 1998, 2001.

27

Kelly, 2020a, 2020b.

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    More recently, Hayden analyzed a new set of ethnographic data “to develop criteria to interpret what was being recorded, in what social contexts, and why.”28 He suggested that decoration might be differentiated from notation through several identifying traits: It fills an entire dimension of the artifact, like its length; its elements are repeated, uniform, and evenly spaced; it is used on objects with social value; it is sized for easy visibility; it is often delimited by a border design; its disposition across the artifactual surface suggests planning; and its patterning is not individually structured or distinctive.29 He further suggested that utilitarian notches, like those made for musical rasps, are deeply incised, and the artifacts themselves will show evidence of wear or curation; in contrast, personal notations are lightly incised on scrap materials.30 Hayden distinguished representational marks from the notations used to express numbers related to debts, calendars, and gambling: Each representation mark expresses “a specific person or item rather than a number count”; these marks tend to be deeply incised on pieces of wood that are specially shaped and decorated.31 Finally, he noted that undifferentiated/ungrouped marks are used for things like calendrical purposes, single transactions or debts, and specific commodities; undifferentiated/grouped marks are used to represent different events or commodities; and differentiated marks are used to express information related to the type of event or commodity.32 Hayden’s analysis is a significant advance in the use of cultural analogy for interpreting Palaeolithic artifacts, particularly in the criteria proposed for distinguishing decorative and utilitarian marks from notational ones. While on the right track, the ability to distinguish numbers from among other notations remains limited because the available ethnographic data are so woefully under-described. Further, the only characteristic of the number system considered was extent, as imperfectly measured by highest number counted

28 31

Hayden, 2021, p. 2. Hayden, 2021, p. 10.

29

Hayden, 2021. Hayden, 2021.

30

Hayden, 2021, p. 5.

32

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     (the limiting number of the linguistic typologist Joseph Greenberg33); this means that we still lack insight into whether the notations used for numbers reflect the numerical organization or other material devices like the Pomo sticks. As Hayden also noted, his results “should be viewed as exploratory rather than definitive. Additional data are certainly required to validate the propositions in this analysis.”34

  :     According to the linguistic anthropologist Piers Kelly,

Message sticks are tools of graphic communication, once used across the Australian continent. While their styles vary, a typical message stick is a flattened or cylindrical length of wood with motifs engraved on all sides. Carried by special messengers over long distances, their motifs were intended to complement a verbally produced communication such as an invitation, a declaration of war, or news of a death. (Kelly, 2020b, p. 133)

As described by the nineteenth-century Australian settler Edward Micklethwaite Curr,

The bearer of an important communication from one party to another often carries a message-stick with him, the notches and lines on which he refers to whilst delivering his message. This custom . . . prevails from the north coast to the south. . . . The extent to which it prevails in Australia, at all events, shows the custom to be one of very ancient date. (Curr, 1886b, pp. 183–184)

Message sticks functioned as “tokens of authority, mnemonic prompts and explanatory diagrams” and communicated purposes like “consolidating alliances, declaring war, announcing ceremonial activities or coordinating population movements” over long distances.35 33

Greenberg, 1978.

34

Hayden, 2021, p. 3.

35

Kelly, 2020b, pp. 133–134.

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    All of their elements were potentially meaningful, not just the motifs inscribed on the surfaces but also the size, shape, and type of wood of the stick itself.36 The messages were primarily verbal, delivered by messengers who both carried the sticks and interpreted their meaning. Designs were also recognized for their cultural affiliations, so a messenger carrying one might be afforded safe passage through a particular territory or his message given careful attention by its intended recipient;37 the marks were considered to be a “guaranty of the messenger, the same as a ring with us in former times.”38 From a numerical point of view, the message sticks are interesting and informative because they were made by societies whose number systems counted no higher than three or four, a restricted extent that is still true today.39 The message sticks were often inscribed with the kind of undifferentiated linear marks or notches that would most likely be interpreted as numerical, if they were found on a prehistoric artifact rather than one from the nineteenth century; as Hayden noted, many are “remarkably similar” to Upper Palaeolithic artifacts identified as notational.40 And the undifferentiated linear marks on message sticks often did not have numerical meaning.41 For example, a flat piece of wood about five inches in length that had been inscribed with both straight and curved lines was interpreted as follows: “‘My wife has been stolen; we shall have to fight – bring your spears and boomerangs.’ The straight lines, it was explained, meant spears, and the curved ones boomerangs; but the stealing of the wife seems to have been left to the messenger to tell.”42 Fig. 11.2 shows three Australian message sticks from the nineteenth century. While examples are often much more ornate, these were chosen because each has seven linear notches. As interpreted by culturally knowledgeable informants – including both aboriginal individuals and European settlers familiar with aboriginal culture – seven

36 39 42

Kelly, 2020b, p. 134. Bowern & Zentz, 2012. Curr, 1886c, p. 253.

37

38 Kelly, 2020b. Curr, 1886a, p. 427. 40 41 Hayden, 2021, p. 5. Kelly, 2020a, 2020b.

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    

. . Three Australian message sticks from the nineteenth century. (Top) Seven notches that meant the number of men killed in battle. (Middle) Seven notches that meant a campsite had been abandoned and was laced with hidden poisoned bones. (Bottom) Seven notches, along with other marks, that carried a message about a wife’s death. Images from Carroll (1896, Fig. 3, p. 10 and Fig. 5, p. 11) and Howitt (1889, Fig. 7, pl. XIV).

notches did not necessarily mean seven of something. Notches could be accumulated to aid the later recall of a series of items.43 This appears to have been the case for the artifact uppermost in the figure. Each notch represented an enemy killed in battle, perhaps prompting the recall of the battles or the circumstances where specific events took place; the other side of the artifact was inscribed with eight notches, each intended to prompt the recall of a particular individual to be killed at a future date.44 The notches on the middle and bottom artifacts had conventional meanings, rather than numerical ones. The middle artifact was intended to warn allies that a campsite had been

43

Roth, 1897; Howitt, 1889.

44

Carroll, 1896, pp. 10–11.

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    abandoned and laced with hidden poisoned bones,45 while the bottom artifact carried the message that a man’s wife had died.46 The message sticks challenge the idea that undifferentiated linear marks necessarily have numerical meaning, since they functioned mnemonically or had conventional or narrative meanings. Nor do the number of marks have any relation to the associated number system. Australian languages typically count no higher than three or four,47 but it is not uncommon for message sticks to have upward of thirty notches.48 And rather than meaning the lunar cycle, nearly thirty notches might mean something along the lines of inviting “all suitable people” (not “twenty-eight people”) for a public ceremony.49

      Table 11.1 lists some of the best-known artifacts thought perhaps to represent prehistoric counting in Africa, the Near East, and Europe. Remember that words for numbers up to five may have been available for more than 100,000 years,50 and that higher numbers most likely emerge under socio-material conditions that are unambiguously present by 30,000 years ago. This means that Palaeolithic artifacts are unlikely have been produced in an absence of numbers, and some of them might have been used by societies with highly elaborated numbers. There is also no particular requirement for Palaeolithic marks to have represented numerical information at all, given the nonnumerical nature of the Australian message sticks. Most of the artifacts in the table are incised bones, but there are also strung beads, hand stencils, and clay tokens, as well as the world’s first-known unambiguous numbers, the clay impressions from Mesopotamia. We will start with the first unambiguous numbers because they illustrate how a numerical notation can be identified using

45 48 49

46 47 Carroll, 1896, p. 11. Howitt, 1889, p. 327. Bowern & Zentz, 2012. E.g., Howitt, 1889, Figs 1, 9, 10, 11, and 12; Hamlyn-Harris, 1918, Figs 8 and 27. 50 Hayden, 2021. Pagel et al., 2013; Pagel & Meade, 2017; Calude, 2021.

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

https://doi.org/10.1017/9781009361262.013 Published online by Cambridge University Press

Table 11.1 Prehistoric artifacts with possible numerical intent, purpose, or meaning. A version was previously published in Overmann (2019b, Table 8.1, p. 147).

Europe

Artifact

Source

Age

References

Hand stencils

Cosquer Cave, France Gargas, France Les Pradelles, France Abri Cellier, France

27 KYA

Abri Blanchard, France

28 KYA

Dolní Věstonice, Czech Republic Grotte du Taï, France

26 KYA

Rouillon, 2006; Overmann, 2014 Leroi-Gourhan, 1967 D’Errico et al., 2018 White & Knecht, 1992; White et al., 2018 Marshack, 1991b; JèguesWolkiewiez, 2005 Emmerling et al., 1993; Oliva, 2014 Marshack, 1973, 1991a, 1991b

Nesher Ramla, Israel Ksar’Aqil, Lebanon Various locations throughout the Mesopotamian Plain

120 KYA “Upper Palaeolithic” 10th/9th–1st mil. BCE (uncertain before mid4th mil.)

Worked bones

Near East

Worked bones Tokens

72–60 KYA 33–28 KYA

12 KYA

Prévost et al., 2021 Tixier, 1974 Broman, 1958; Oppenheim, 1959; Amiet, 1966, 1972; Schmandt-Besserat, 1992a; Moore & Tangye, 2000; MacGinnis et al., 2014; Overmann, 2019b

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Impressions & notations Africa

Beads Worked bones

Mid-4th mil. and thereafter

Nissen et al., 1993; Chrisomalis, 2010; Overmann, 2019b

Blombos Cave, South Africa Border Cave, South Africa

75 KYA

Ishango, Democratic Republic of the Congo

25–21 KYA

D’Errico et al., 2005; Coolidge & Wynn, 2011 Beaumont, 1973; d’Errico et al., 2012, 2018 de Heinzelin, 1962; Brooks & Smith, 1987; Pletser & Huylebrouck, 1999; ADIARBINS, 2018

42 KYA



     paleographic techniques. The numerical meaning of the Mesopotamian impressions is attested by their correspondences of shapes, sizes, and quantities with later proto-cuneiform notations and cuneiform numbers, which are unambiguously numerical.51 That is, the relations in the proto-cuneiform notations were organized by means of bundling, in which six or ten notations of one shape and size were equivalent to one notation with a different shape and size. With multiple artifacts available for statistical analysis, these relations could be reconstructed through their unbundled maximums.52 The availability of multiple artifacts with proto-cuneiform notations, their correspondences of shape, size, and quantity with the earlier impressions, and the ability of the impressions to display unbundled maximums and other numerical relations were key to identifying the impressions as numerical. Some of the earliest numerical impressions are also found with small geometric objects made of clay, “tokens” used for counting before the practice of impressing them emerged. These assemblages correspond well enough in their shapes, sizes, and quantities to verify the numerical purpose of tokens. While their correspondences with later unambiguous forms attest to tokens’ use as numerical counters in the mid-fourth millennium BCE, this same certainty does not obtain for small clay objects that are not accompanied by numerical impressions, some of which have been found as early as the tenth or ninth millennia BCE.53 And there are good reasons to be skeptical about interpreting unaccompanied clay objects as numerical. First, there has been a tendency to label any small clay object as a “token,” regardless of its actual purpose.54 Second, find contexts and prevalence rates do not support numerical interpretations; third, small clay objects might just as plausibly have functioned as funerary offerings, game pieces, children’s toys, or tools; and fourth, there are no reliable

51 53 54

52 Amiet, 1966; Nissen et al., 1993. Nissen et al., 1993. Schmandt-Besserat, 1992a; Moore & Tangye, 2000; Overmann, 2016b. Schmandt-Besserat, 1992a, 1992b.

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    methods or criteria for distinguishing numerical intent from other purposes.55 Tokens thus show that the paleographic comparative technique can be extended to artifacts under specific circumstances, and that archaeological techniques by themselves find it difficult to establish the numerical meaning of artifacts. We have already covered most of the techniques and issues associated with the artifacts in Table 11.1: Impressions and notations are understood through paleographic techniques, as are the tokens found in assemblage with corresponding impressions. There are no conclusive techniques or criteria for establishing the numerical meaning of hand stencils, beads, or unaccompanied tokens. Marks on bones and other hard surfaces undergo microscopic analyses and experimental/ethnographic comparisons; these techniques have found it difficult to establish the meaning of undifferentiated linear marks, numerical or otherwise, partly because these marks do not comprise a code of any kind and partly because the artifacts tend to be one-offs. What we turn to now are the other kinds of considerations that might help us interpret artifacts with marks: functional considerations of size and quantity and their implications for use in numbers.

  Consider the Nesher Ramla artifact, a piece of aurochs bone dated to 120,000 years ago incised with six linear marks (Fig. 11.3).56 Quantity, as we saw with the Australian message sticks, is not necessarily informative about the extent of a number system, so the fact that there are six incisions on the piece of bone is not necessarily meaningful in this regard. The marks have minor differences in their morphology and spatial distribution. One is bifurcated at the top (Fig. 11.3c), and there is a slight suggestion of a second bifurcation on a second mark (Fig. 11.3a). The size of these features – about a centimeter (0.4 inch) for the longer of the two bifurcations – makes it 55 56

Shendge, 1983; Oates, 1993; Zimansky, 1993; Friberg, 1994; Englund, 1998b. Prévost et al., 2021.

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    

. . Engraved aurochs bone from Nesher Ramla, Israel, dated to 120,000 years ago. The artifact is 8.1 centimeters (3.2 inches) measured horizontally and 5.3 centimeters (2.1 inches) measured vertically; the incisions are 3.8–4.2 centimeters (1.5–1.7 inches) in length. Adapted from Prévost et al. (2021, Fig. 3a, p. 5). Image © Elsevier and used with permission.

difficult to argue that they represented intentional encoding. There are also slight differences in the length and spacing of the linear marks. Again, the largest difference in either dimension is less than a third of an inch, a size that argues against the differences being significant. As microscopic analyses and experimental comparisons suggest that the marks were made intentionally, the authors argue that the incisions were not doodles and most likely had a communicative purpose or a symbolic meaning.57 The archaeologist Derek Hodgson has challenged this on the grounds that an alternative explanation, visual sensitivity to pattern, is available; on this interpretation, the marks may have had a proto-aesthetic value.58 As Prévost and her 57

Prévost et al., 2021, p. 12.

58

Hodgson, 2022; Prévost & Zaidner, 2022.

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    colleagues also admit, the remains of flesh obscure the cutmarks made on bony surfaces during butchery, and in the case of this artifact, “There is no direct evidence that the bone was cleaned before the application of the incisions.”59 This circumstance argues against the incisions having been made for their visual appeal (Hodgson’s argument) or having served a communicative or symbolic purpose (Prévost and colleagues’ claim). The Lebombo bone from Border Cave, South Africa (Fig. 11.1) is dated to 42,000 years ago; it has twenty-nine incised notches that are roughly the same in length, spacing, and orientation, and its wear, a glossy polish, shows that it was curated for a long time.60 The artifact was said to resemble wooden sticks used as calendars by “some Bushman clans in South West Africa,”61 a claim that has been widely repeated. However, there are no photos and descriptions of the calendar sticks purportedly used by the San people for comparison. Further, culturally linking extant peoples to those inhabiting the same region some tens of thousands of years earlier is problematic, and attempts to construe cultural continuity over such a long period of time have been met with criticism,62 as the “weight of modern evidence is against the notion that contemporary human cultures can be tracked backwards into the Pleistocene.”63 While this is true for material culture in general, it is also true that numerical traditions can span impressive amounts of time, like the Palaeolithic roots of our Western numbers. The Lebombo bone is incomplete, so it potentially held more than twenty-nine marks when it was whole. The artifact is still no exception to the tendency to interpret this number as comprising a lunar cycle, so let us say, for the sake of argument, that the artifact was just that. How exactly would it have been used? While the object is worn in a way that shows it was curated, retention and handling

59 62

63

60 61 Prévost et al., 2021, p. 7. D’Errico et al., 2012. Beaumont, 1973, p. 44. See d’Errico et al., 2012; criticisms in Mitchell, 2012, Pargeter, 2014, and Pargeter et al., 2016a; debate in Barnard, 2016, Denbow, 2016, d’Errico et al., 2016, and Wobst, 2016; and final reply in Pargeter et al., 2016b. Pargeter et al., 2016a, p. 1087.

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     differ from functional use. We might predict that such a device would need to be incremented, one notch per day, to show where a particular day fell with respect to the whole cycle; otherwise, what would be the point? Yet there are no indications of the artifact having been used with twine or coloration, the kinds of things we might expect to be used in repetitively incrementing the notches. Both the artifact and its marks are relatively small, so keeping track of incrementations might have been difficult in any case. What would be needed to support the argument that the artifact was used as a lunar calendar would be an ethnographic attestation (with detailed descriptions of use, photographs or drawings, and dimensions) of an object of about the same size used for that purpose, and so far, none has been documented. A North American calendar stick, in comparison, was reported to be 132.0 centimeters long (52.0 inches).64 Subtracting the unnotched portion (about 23 percent) and dividing the result by the six lunar months incised on each edge yields 16.9 centimeters (6.7 inches) per lunar month, more than double that of the Lebombo artifact (7.6 centimeters or 3.0 inches). The notations on the North American artifact are also larger, making them easier to see and use. Size would also be a consideration in the alternative use as a musical rasp proposed by Hayden, based on the deep incisions and heavy wear,65 as the volume of sound produced by contemporary examples is also related to size (North American and African examples held in the collection of Harvard’s Peabody Museum of Archaeology and Ethnology range from 32.0 to 80.5 cm, four to more than ten times the size of the Lebombo bone). The Les Pradelles bone (Fig. 11.4) is dated to 72,000–60,000 years ago and is attributed to Neandertals.66 It contains a row of nine large incisions about 0.45–0.60 centimeters (0.18–0.24 inches) in length and spaced an average of 0.5 centimeters apart. Below the main row are two sets of four partly-overlapping incisions 0.05–0.20 centimeters (0.02–0.08 inches) in length and spaced less 64

Marshack, 1985.

65

Hayden, 2021.

66

D’Errico et al., 2018.

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   

. . Engraved hyena femur from Mousterian site at Les Pradelles, France, dated to 72,000–60,000 years ago. The artifact is 5.33 centimeters (2.1 inches) measured horizontally and 1.97 centimeters (0.78 inches) vertically; the large incisions are 0.45–0.60 centimeters (0.18–0.24 inches) in length, while the small ones highlighted with a circle are only 0.05–0.20 centimeters (0.02–0.08 inches). Adapted from d’Errico et al. (2018, Fig. 1, p. 4).

than 0.05 centimeters apart (these have been circled in the figure to highlight them); to put this in better perspective, the longest measurement, 0.2 centimeters, is about 1/16th of an inch. Their placement beneath and between the second, third, and fourth incisions of the main row might suggest counting by fives. The numerical interpretation67 is problematic for several reasons. First, the lower marks are minute and thus quite difficult to see. Conceivably, the need to recount them would potentially be mitigated by their alignment underneath the larger marks, similar to the way a horizontal line in a tally (卌) mitigates the need to count the vertical lines. While they have been interpreted as intentional, their minute size suggests that they may have represented decoration, as the two groups are positioned and sized to balance one another. Second, interpreting adjacent small and large marks as comprising groups of five leaves in doubt the interpretation of the remainder of the main row of marks.

67

D’Errico et al., 2018; also see comments in Prévost et al., 2021.

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     And if the larger marks were intended to indicate groups of five, it is a good question why there are only two intervening small groups of four. Significantly, the Neandertal material record does not contain many artifacts inscribed with linear marks, which are relatively common for H. sapiens. This is consistent with the differences in skull morphology and implications for cognitive functions like creativity discussed in Chapter 3. Simply, the one-off artifact with linear marks that is situated in a cultural context containing lots of other artifacts with linear marks must be treated differently than one situated where such artifacts are rare or absent. The former has a greater likelihood of having had a numerical purpose, the latter of having had a nonnumerical function or being an instance of cultural transfer (though in this case, cultural transfer from H. sapiens is unlikely because the artifact predates their arrival in Europe). Another interpretation, based on cultural analogy, is a series of marks used in wayfinding, with the smaller marks intended to identify one particular waypoint as perhaps including a feature like a waterfall (e.g., the Barasana map of river rapids as a series of lines representing waypoints, some decorated).68 The degree to which contemporary H. sapiens provide relevant cultural analogies for prehistoric cousin species is unknown. Ultimately, this artifact, like all one-offs, remains provocative, especially in its association with the Neandertals, but it does not conclusively identify prehistoric counting. The Taï plaque is relatively recent at 12,000 years (Fig. 11.5). It contains over a thousand marks, each only 0.1–0.2 centimeters (0.04–0.08 inches) in length (as extrapolated from the artifact’s maximum length), thought to have been accumulated over a period of several years.69 The marks represent a quantity that is not subitizable, a size that is not easy to see or use in any quantity, and a period of time that significantly challenges human memory. To modern eyes, 68

Hugh-Jones, 2016.

69

Marshack, 1973, 1991a, 1991b.

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   

. . Two views of an engraved bovine rib fragment from the Grotte du Taï, France, dated to 12,000 years ago. The artifact is 8.7 centimeters (3.4 inches) measured horizontally and 2.7 centimeters (1.1 inches) vertically; the incisions, over a thousand in all, are 0.1–0.2 centimeters (0.04–0.08 inches) in length. Adapted from Marshack (1973, Fig. 1, p. 13).

the Taï plaque looks like it might have had a numerical purpose, and it certainly represents accumulation. Accumulation is distinct from a concept of number; its value may not have arisen from its representing a number per se, but rather, in the simple fact that it accumulated. In fact, large quantities of undifferentiated marks might imply a number system that is not highly elaborated. This is because elaborated numbers come with expectations and constraints that restrict the utility of undifferentiated marks in large quantities. Consider how inaccessible the marks on the Taï plaque, taken as numerical, would be to us, Western people enculturated into a highly elaborated system of numbers. We expect numerical information to be accessible because our numerical representations are accessible, a property they achieve by consolidating and concisely representing the units, tens,

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     and hundreds. Since the information on the Taï plaque is not concise, we do not find it accessible. Our expectation interferes with our ability to access the information being represented. But the lack of accessibility may be a problem only where accessibility is expected. If there is no such expectation, the constraint disappears. A society like the Pomo would surely find it difficult to use as well, not just because their concise material recording would make them expect accessibility but also because their system of counting involved replacing quantities of smaller bundles with single larger units; this differs substantially from the incremental sequence required to count from one to a thousand one by one. By this logic, the number system associated with the Taï plaque was unlikely to have been highly elaborated. Hayden argues that the quantity of notations is “difficult to conceive of as existing without complex numbering systems of comparable magnitudes,”70 and undoubtedly, the specialization implied by having someone spend several years to accumulate the marks argues in favor of complex socio-material conditions. Nonetheless, the conditions do not entail that numbers have elaborated, only that they have the potential to do so. The debate will hopefully be settled through further developments of contemporary models. Complicating our understanding of how undifferentiated linear marks should be interpreted vis-à-vis societal numeracy is the Ishango bone, an artifact that is 25,000 years old (Fig. 11.6).71 Its three rows of linear marks are grouped by differences of spacing into mathematically suggestive quantities. Row (a) contains the prime numbers between ten and twenty in ascending order; Row (b) contains the numbers produced by adding and subtracting one from ten and twenty; and Row (c) contains halves and doubles, though not consistently.72 The seeming impossibility of producing such sequences by

70 71 72

Hayden, 2021, p. 16. De Heinzelin, 1962; Brooks & Smith, 1987; Brooks et al., 1995. De Heinzelin, 1962.

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   

. .. Engraved bone from Ishango, Democratic Republic of the Congo, dated to 25,000–21,000 years ago. The surface of the bone has been worked to such an extent that the species cannot be identified (ADIARBINS, 2015). The artifact is 10.0 centimeters (3.9 inches) long and 1.6 centimeters (0.6 inches) wide; the incisions are 0.20–1.30 centimeters (0.08–0.51 inches) in length. Adapted from de Heinzelin (1962, left image, p. 114).

chance has fueled speculation about mathematics having deep African roots, though this interpretation has not been without controversy.73 The device also 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 in any significant detail, presumably because its ninety marks are not mathematically interesting.74 While mathematically suggestive, the marks are generally consistent with the restricted range of numbers, since only one group – the twenty-one marks in Row (b) – breaks the upper limit of twenty. If they were numbers and numerical relations, they suggest an impressive degree of insight, something that typically requires the use of a manipulable technology. The marks are not bundled, which means

73

74

Huylebrouck, 1996; Pletser & Huylebrouck, 1999, 2015; Keller, 2010; Evans, 2014, p. 4. ADIA-RBINS, 2018.

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     the numerical information is presented without the conciseness that facilitates accessibility and intelligibility. The combination of conceptual complexity and notational clumsiness is difficult to explain. One potential explanation is that the lack of conciseness is not as great an impediment to mathematical conceptualization as we Western folks might think. Another is that mathematical concepts could have been worked out with a manipulable technology, to be later recorded using an unconcise notation. While this might eventually motivate the development of conventional signs for bundled values, the combination might also imply a numerical tradition in which manipulation acted as the central organizing principle. As will be discussed in Chapter 12, some of the number systems in West Africa may have originated as this type. Many more artifacts could be discussed, but the ones shown are hopefully sufficient to establish three things. First, we really do not know when or how numbers started, and the archaeological record currently sheds little light on these questions. Second, archaeological techniques are often frustratingly inconclusive regarding the interpretation of artifacts as numerical, especially for undifferentiated linear marks, and this is a matter that cultural analogies have significant potential to help resolve. Third, function ought to be considered when interpreting artifacts, and the implications of characteristics like size, quantity, and use are often ignored. This too is potentially addressable through cultural analogy.

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 Devices That Accumulate and Group

We turn now to technologies that can be moved and rearranged, like pebbles and cowrie shells. These material forms and practices both accumulate and group (Fig. 12.1). Accumulation adds like the tally does: one, two, three, four, five, and so on; adjacent markers differ by one. Grouping makes numerical information more concise: One kind of pebble – perhaps one with a certain size, shape, or color – might represent a group of ten, and a pebble with a different appearance might represent one. This reduces the number of pebbles by replacing multiple units of lower value with one of higher value. Alternatively, pebbles might take their value from their spatial placement – their literal place value as units or tens. This reduces the total number of elements needed because ten is represented by a single pebble in the tens place. These strategies bring new relations into the number system, as for example, ten of a lower value make one of the next higher value. Rearranging loose elements like pebbles has the potential to yield even more relations: For example, arranging a group of ten pebbles into two smaller groups produces two groups of five, and arranging a group of ten pebbles by twos produces five such pairs, potentially occasioning ideas of relations between ten, five, and two. Using a material form that is manipulable and grouped will also affect how numbers are conceptualized: Rather than things that are ordered and related merely as next and one-more, as in the tally, they become things that can be moved around, and which also have the relations enabled by grouping and manipulation. The fingers are naturally grouped, so they too have the potential to yield concepts of grouping. For example, in digit-tally systems, numbers are naturally grouped into four groups of five – the first hand, 

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    

. . Devices that accumulate and group. (Left) Counting board (showing the number 9999). Each row accumulates a maximum of nine markers (pebbles, calculi, jetons, etc.), with a tenth marker occasioning replacement of nine in the lower row by one in the next higher row. For grouping, values are indicated and separated by place, with each higher row being ten times the value of the lower row. (Right) Notional decimal abacus (showing the number 1). For accumulation, each column accumulates a maximum of nine beads by sliding them to the top, with a tenth marker occasioning their being slid to the bottom and one bead in the next row to the left being slid to the top. For grouping, values are again indicated by place, with each column to the left being ten times the value of the column to the right. Actual decimal abacuses typically have an upper area with one or two rows whose beads represent five, creating exchange values (at the fifth bead(s)) within each column. Other variants include the Roman abacus, whose fractions were duodecimal (base 12), and the nepohualtzintzin, a vigesimal abacus said to have been used in pre-Hispanic America (Murray, 1930; Micelli & Crespo, 2012). Image by the author.

the second hand, the first foot, and the second foot – that cumulatively yield a person’s worth of digits, or twenty. This grouping can manifest in language as a sub-base, with numbers formed from the four multiples of five (for example, fourteen might be one from fifteen). Groups are sequentially added in value, not positionally valued. Number systems in which the hands have different positional values (e.g., units on the left hand, tens on the right) are not only rare, they

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      are also associated with highly elaborated finger-counting, like the system documented by the Venerable Bede1 (Chapter 9). In addition, as integral parts of the body, the fingers and toes cannot be manipulated in the same way that loose objects can. Accordingly, while human digits are a grouped technology, they will not be discussed further in this chapter. The category of devices that accumulate and group encompasses a wide range of material forms and practices. Some of these forms and practices are relatively simple and ad hoc, like picking up a stone for every ten coconuts collected, a practice recorded in the Marshall Islands of Micronesia.2 Some are formal devices: Counting boards were used widely throughout the West and in Mesoamerica from ancient to relatively recent times,3 and fixed-bead abacuses include the Chinese suanpan, Japanese soroban, and Roman abacus. The earliest known counting board, the Salamis tablet, is dated to 300 BCE,4 though counting boards were likely used in Mesopotamia during the Old Babylonian period (1900–1600 BCE), as inferred from the use of linear columns and certain types of calculation errors in texts.5 While many of these devices indicate differences of value by physically separating counters that are otherwise identical in form, some of them encode values through differences of shape and size, like the Mesopotamian tokens.6 Some of them sort the objects being counted, again separating them by physical place, a practice found in Polynesia

and

in

Papua

New

Guinea,7

and

some

count

collaboratively, with different individuals keeping track of the units, tens, and hundreds, a practice documented in Oceania and Africa.8 In this chapter, we will focus on three technologies associated with highly elaborated number systems: the Inka counting board, the

1 3 4 6 7 8

2 Bede, 1999. Krämer, 1906, p. 438. Pullan, 1968; Stone, 1972; Evans, 1977; Sugden, 1981; Reynolds, 1993. 5 Kubitschek, 1900. Ifrah, 1981; Menninger, 1992; Proust, 2000; Woods, 2017. Schmandt-Besserat, 1992a; Nissen et al., 1993; Overmann, 2016b, 2019b. Overmann, 2020c, 2021b, 2021c. Schrumpf, 1862; Collocott, 1925, 1927; Churchward, 1941.

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     sorting practices of Oceania, and the cowrie counting systems of West Africa. These cases will illustrate particular methods, while highlighting their influence on the resultant numerical concepts and language; this will emphasize the main theme of the book, the idea that numerical organization and structure result from the material forms used and how they are used. These cases will also serve to reiterate some of the basic principles whereby numbers emerge and become elaborated through the use of material forms: Accumulation emerges before manipulation does, previous material forms influence the organization and structure of subsequent ones, and multiple material forms work together as a coherent system. We will also see that familiarity with Western numbers can influence how we view numbers in other systems, especially ones that are highly elaborated but which do not incorporate written notations, with the result that we can misinterpret and misunderstand them, sometimes badly.

    The numerical device comprised of multiple knotted strings known as the khipu (Fig. 12.2, left) is justly famous, even if the way it works as writing, especially in its nonnumerical component, still is not fully

. . Khipu and yupana. (Left) Khipu (or quipu), a device for representing numbers used by the Inka (also see Fig. 13.3). Image © Trustees of the British Museum and used with permission. (Right) Drawing of three-dimensional device possibly used for calculating (yupana), as a game board, or to represent sacred landscape in Peru and/or Ecuador. Compare with Fig. 7.2, Guamán Poma’s flat yupana. See Gonzalez Suarez (1892, Plate III). Image in the public domain.

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      understood (something examined in the next chapter). Less well known is the khipu’s limitation as a calculating device, the lack of manipulability inherent in tying, untying, and retying knots. And calculational manipulability was most definitely needed. The Inka number system was vital to administering the largest bureaucratic state in pre-Columbian America, an empire that emerged in the early 1200s, flourished between 1438 and 1532, and had been fully conquered by Spain by 1572. Given this complex socio-material context, it is not surprising to find that these numbers were highly elaborated and counted into the millions.9 While the khipu was an effective way to record, store, and communicate numerical information across time and distance, as a fixed form, it was thoroughly unsuitable for manipulating that information. Calculation was performed instead on a counting board or yupana, a name derived from the Quechuan word yupay, which means to count. These devices are thought to have had various forms – flat (Fig. 7.2) and three-dimensional (Fig. 12.2, right) – though it is unclear how the threedimensional ones were used in calculating,10 or even whether they were, since they might have served as game boards or represented sacred landscape.11 Our focus here is the flat yupana documented sometime between 1583 and 1615 by a Quechua nobleman, Felipe Guamán Poma de Ayala.12 His book, El Primer Nueva Corónica y Buen Gobierno (The First New Chronicle and Good Government), recounted the history of the Andes and detailed various aspects of Inka society, including its elaborate bureaucracy. Guamán Poma described Tahuantinsuyo, the chief accountant of the empire, as follows:

[Tahuantinsuyo] was very capable. To test his skill, the [Inka King] had him count and correct the number of Indians in this kingdom. He kept track of the amount of wool of the deer called taruca, that of the Indians, and the

9 10 11

Guamán Poma, 1615, p. 361. Pullan, 1968; Sugden, 1981; Aitken-Soux & Ccama, 1990. 12 Gonzalez Suarez, 1892; Smith, 1977. Guamán Poma, 1615.

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     food called quinoa.[*] He counted the quinoa and the Indians. He was very capable, as if he used paper and ink. . . . They count on boards. They count by numbers from one hundred thousand, ten thousand, one hundred, ten until they reach one. They keep track of everything that happens in this kingdom, celebrations, Sundays [feast days], months and years. In each city, town and village of Indians there were accountants and treasurers. (Guamán Poma, 1615, p. 361; as translated by Hamilton, 2009, pp. 286–289) [*]

Hamilton found this passage difficult to understand and noted that the taruca was used for meat, not wool (p. 325).

Some translations of Guamán Poma’s archaic, idiomatic Spanish mention quinoa not as an object being enumerated, but as comprising the counters used with the yupana.13 Like the idea that the khipu can be used for calculating, this too misunderstands the practicalities of manipulating material forms in numbers, since quinoa seeds are about 0.2 centimeters (or 1/16th of an inch) in diameter – about the same size as some of the prehistoric tally marks noted in the last chapter as being unusable because they were particularly small. Their minute size means that quinoa seeds cannot be easily seen or grasped, but in a counting arrangement, they would be easily disturbed by movement or breeze. The 1586 account of the Jesuit missionary José de Acosta, La Historia Natural y Moral de Las Indias (The Natural and Moral History of the Indies), suggests that kernels of maize were used instead.14 Dried kernels of maize, it should be noted, are about five times the size of quinoa seeds.15 None of the known historical accounts describe how the yupana was used. Counting boards were quite familiar to Western observers at the time, since they were the dominant form used for calculating in Europe, though algorithms for calculating with notations were beginning to gain in popularity (Fig. 1.4).16 Further, because both Inka and

13 15

14 E.g., Leonard & Shakiban, 2011, p. 84; Tun, 2014, p. 2. Acosta, 1589, p. 411. 16 Sangamithra et al., 2016. Stone, 1972; Evans, 1977; Reynolds, 1993.

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      Western counting boards were used with decimal numbers that could count into the millions, the Spanish may have assumed that Inka numbers were essentially the same as Western numbers, and indeed, the two systems closely resembled one another in terms of their organization, structure, and extent. Thus, it is possible that no one thought to record the details of calculation, finding them unremarkable in and of themselves, as well as thoroughly subsumed by the mysterious khipu string-writing used to record them. Another interpretation is offered by the Italian researcher Cinzia Florio, who sees the yupana as involving the kind of bundling/debundling operations used with the Mesopotamian tokens, rather than the place value used in Western counting boards.17 She speculates that kernels of maize in different colors were used to represent the various multiples of ten, up to the tens of thousands occasionally represented in khipus.18 Bundling/debundling is a different logic than the spatial placement used on a Western counting board or abacus (Fig. 12.1); the speed at which the exchanges were made might have resulted in the confusion articulated by Father Acosta, who wrote in 1589,

It is an enchanting thing to see another kind of quipu which they work with grains of maize, since for a very difficult count, for which a good calculator would need pen and ink . . . the Indians will take their grains [of maize]. They will place one here, three there, eight I do not know where. They will move one grain from here, bring three from there, and actually finish with their count absolutely correct without the smallest error. And they are much better at calculating what every one should pay or give than we with pen and ink. (Acosta, 1589, Book 6, p. 411, as translated)

Adding to the confusion on how the Inka used the counting board is the fact that no one is entirely sure what one looked like,

17

Florio, 2009.

18

Chrisomalis, 2010.

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     and as noted earlier, there are two candidates. Three-dimensional artifacts (Fig. 12.2, right) have been found, though not in Peru. The flat version (Fig. 7.2) comes from Guamán Poma’s 1615 book; it is also suggested by finds like the series of circular holes impressed into the floor at an archaeological site used as an administrative center.19 To date, neither form has been successfully decoded, the former because its logic differs from the counting boards used in the West, the latter because there are multiple ways to interpret it. Most attempts to interpret the black-and-white counters on Guamán Poma’s flat board have assumed that Inka numbers were a positional system, identical in this regard to Western numbers. In a positional system, 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 row of a counting board or the ones, tens, or hundreds column of an abacus. On a khipu, a knot’s physical placement along a string served the same purpose: The knots at the end of a string represented the units, the next higher cluster of knots the tens, and the next higher cluster the hundreds and so on, as will be discussed in more detail in Chapter 13. Thus, in many interpretations, the rows on Guamán Poma’s yupana have been given positional values as the increasing powers of ten. With Western numbers on a Western counting board, each counter is valued at one, as weighed by the value of the row in which it is placed (one unit, one ten, one hundred, and so on; the second counter placed in that row means two units, two tens or twenty, two hundred, etc.). Despite the prediction that the Inka decimal numbers might follow this same decimal structure, not all of the attempts to decode the yupana have been valued in this way. The first attempt (Fig. 12.3a) valued the columns (from left to right) as 1, 5, 15, and 30; this calculation yielded the number 408,257.20 While the column valuation was not explained, it did not correspond to either khipu organization or Inka numbers. The second attempt (Fig. 12.3b) valued each of the columns as 1, as consistent with Western numbers and 19

Barraza Lescano et al., 2022.

20

Wassén, 1931.

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     

. . Four interpretations of Guamán Poma’s yupana. (a) Wassén; (b) Radicati di Primeglio; (c) de Pasquale; and (d) Florio. The first three assume Inka numbers to be a positional system like Western numbers, while the last assumes them to be an additive system akin to Roman numerals. The latter rearranges the counters and then interprets one black counter as a mistake (indicated by the X superimposed on a white counter in the leftmost column). Image by the author.

counting boards; this yielded the number 53,636.21 While this interpretation did correspond to khipu organization and Inka numbers, the result was mathematically uninteresting: It did not contain any

21

Radicati di Primeglio, 1979.

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     numerical message or algorithm. The third attempt (Fig. 12.3c) interpreted the calculation as an astronomical reckoning, valuing the rows as multiples of 40 and the columns (from right to left) as 1, 2, 3, and 5; this yielded the number 44,139,620.22 Like the first attempt, this one also did not correspond to khipu organization or Inka numbers. More recently, Florio took a different approach (Fig. 12.3d).23 First, she rearranged the counters so that all the white ones were in the top three rows, all the black in the bottom rows. They do not separate cleanly: One black marker remains in the white group, which she argued was a mistake in the drawing. Significantly, she also reinterpreted the notations as an additive system, wherein signs are repeated with the expectation that they will be added. The result is similar to Roman numerals, where XXXII means thirty-two because each X means ten and there are three of them, while each I means one and there are two; similarly, the rightmost column in the yupana is read as thirty-two because the three white counters are worth ten each and the two black counters are worth one each. The result, read from right to left, forms the equation 32  5 = 160. Column 1 (the rightmost) displays 32. Columns 2 and 3 effectively multiply the values in Column 1 by, respectively, two and three, yielding 64 (six white plus four black markers) in Column 2 and 96 (nine white plus six black markers) in Column 3. Finally, Column 4 (the leftmost) adds Columns 2 and 3 to produce 160: fifteen white and ten black markers, given one black marker interpreted as white. The result is the kind of numerical message or algorithm we might predict such a diagram would provide. Why Guamán Poma might have scrambled the counters is unknown, though such practices were not uncommon in historical documents, where they served to make information esoteric, knowable only to initiates. Most importantly, Florio’s interpretation reveals a close reading of the material forms involved. Most scholars have interpreted Inka knots as positional numbers, likely expecting 22

De Pasquale, n.d., 2001.

23

Florio, 2009.

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     

. . Positional (top, Western) and additive (bottom, Roman) numerals and their corresponding (identical) representations in khipu knots. How khipu knots represent numbers is detailed in Chapter 13. Image by the author.

positionality not just because of their familiarity with Western numbers and the tendency to interpret other numbers in terms of them (the “backward appropriation“24 mentioned in Chapter 1), but also because Inka and Western numbers correspond in so many other respects, including the possible concept of zero implied by sections of string left untied to represent the absence of a number. Nonetheless, Florio accurately noted that the identical knots would be equally consistent with both positional and additive numbers (Fig. 12.4), as knots take their value from their physical position on the string in either case. Here we will apply the principle of parsimony, the idea that the simpler explanation – in this case, numbers being additive – is more

24

Rotman, 2000, p. 40.

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     likely to be true; it is also consistent with the finding that additivity emerges earlier than positionality does in notations.25 As further evidence that her interpretation is correct, the calculation and knots directly correspond. And this is exactly what we should expect to find, since the manipulable form would have been influenced by the accumulating form, and both would have developed to work together. That is, any disconnects between the forms used for representing and manipulating would be resolved by changing one or the other in order to make conversions between the two seamless; this is just the principle that people will find ways that are efficient and minimally effortful, and this will influence the form and function of material devices. Now that we have looked at decimal, linear forms of abacus, let us look at another decimal, linear abacus, one that is ephemeral in its form, the sorting practices of Oceania.

      “ ” Given the task of counting a pile of objects – say, a pile of dried beans – Western people tend to count them by sorting them into piles of ten, then combining ten piles of ten as a group of a hundred, and then ten groups of a hundred as a thousand. Yet, this is not an efficient way of counting, as can be seen by thinking about what would be involved in counting something much heavier and bulkier than beans: the coconut. A typical coconut weighs about 1.6 kilograms (roughly 3½ pounds), so ten would weigh about 16 kilograms, a hundred about 160 kilograms. These amounts are already too heavy to shift and combine, and yet they still fall far short of the amounts to which physical counting has been documented in the ethnographic literature. In addition, the method that works with small items like dried beans involves handling each item multiple times; while this is not

25

Chrisomalis, 2010.

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      too costly in terms of physical effort when it comes to beans, it certainly would be with coconuts. Another strategy Western people often try is counting beans one by one, in the process moving them from an uncounted pile to a counted group. Here the problem is that sequential counting – four hundred and eighty-three, four hundred and eighty-four, four hundred and eighty-five – imposes substantial demands on resources of attention and working memory, even without the potential for distractions. Shortening the phrases, perhaps by omitting the repetitive four hundred and eighty part, can help, but the counting itself is still subject to interruption and forgetting. And while the amounts that need to be shifted have been reduced to single beans because no groups of ten or hundred are being created, the requirement to handle the beans still remains. This discussion of potential methods for counting beans and coconuts illustrates some of the challenges, trade-offs, and solutions inherent in counting physical objects. This is important because as numbers emerge, they are tools for counting physical objects. Only later do they become the kind of numerical entities found in elaborated mathematical traditions: abstract and relational and seemingly independent of both the objects being counted and the devices used for counting them. And physical counting is only infrequently part of the quotidian Western lifestyle, as counting is more likely to consist of word problems encountered in arithmetic instruction, where the objects being counted are merely described, or bar-code scanning in supermarkets, where the movement part of physical counting remains but the mental part has been automated. In contrast, when real objects are counted, especially when they are counted often or to significant amounts or both, properties like weight, size, shape, and graspability entail physical and mental considerations that inform both the counting methods employed and the resultant numbers themselves. A fundamental condition of physical counting is that objects must be moved or otherwise handled in order to count them. This is

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     because counting by visual inspection alone is infeasible, for two reasons. First, our attentional capacity cannot keep sufficient track of what has been counted, relative to what needs to be counted; second, at least some of the objects to be counted will likely be hidden beneath or behind others. Thus, there is an implicit need to distinguish the objects that have been counted from the objects remaining to be counted in some way, and separating them spatially by picking them up and moving them is an effective way to differentiate the two. Moving objects involves physical effort – even for small items, if there are enough of them – and it is repetitious, meaning that it is tedious (mentally taxing) and prone to error (the effect of mental taxation). People who need to count physical objects naturally gravitate toward methods that reduce the physical and mental effort involved, experimentation and trial-and-error discovery that is passed on to others through observation and teaching. Reduced physical and mental effort, in turn, makes counting more efficient and more accurate. A particularly effective method of counting is by sorting, which is also called tally-counting. Evidence of tally-counting has been found throughout Polynesia and in Papua New Guinea. It was first encountered by Western explorers when they reached Polynesia in the early 1500s, by which time, the region had been inhabited for hundreds to thousands of years. In Polynesian tally-counting, every tenth item is set aside to represent ten of the items being counted (Fig. 12.5a).26 This creates two piles, those that have been counted and those that are now valued at ten each. If the items now valued at ten each are more than ten in their number, these are counted in the same way, with the result again that there are two piles, those that have been counted and those that are now valued at one hundred each. This process continues until fewer than ten items remain to be counted, and their number represents the most significant digit in counting. For example, in Fig. 12.5a, the first round of counting

26

Overmann, 2020c, 2020a, 2020b, 2021c, 2021b.

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     

. . Polynesian tally-counting for 1234 items in units of (a) singles, (b) pairs, and (c) fours. In each case, counting is decimal (one through ten), but the unit of counting varies from one to four items. As can be seen, fewer movements are required to count multiple items, a significant physical efficiency. Units like hundred were understood as doubled (twohundred-unit) or quadrupled (four-hundred-unit) when the unit of counting was pairs or fours. Throughout Polynesia, remainders were generally rounded off but acknowledged with terms meaning “remainder”; this practice avoided the need to convert between units and partial units when counting by pairs or fours. Image by the author.

creates 123 counters worth one ten-unit each; the second round creates twelve counters each worth one hundred-unit; the third and final round creates a counter worth one thousand-unit. Remainders were acknowledged by terms that meant “remainder”; this practice avoided the need to convert between units and partial units, a complication when the unit of counting was a pair or group of four, described below.

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     Tally-counting or counting by sorting incorporates significant physical and mental efficiencies: No groups with increasing weight and bulk are created, so none need to be shifted and combined; most of the items being counted are handled only once; and counting words are reduced to one to ten. Further, the items being counted anchor the process, not only minimizing demands on attention and working memory, but also creating a material record of the count. Tallycounting creates an ephemeral abacus, ephemeral because the device, the piles of objects that differentiate them as already counted or remaining to be counted, persists only while counting is performed, abacus because the material form, while it exists, instantiates positional value and facilitates the exchange of value between groups. Depending on considerations of weight, size, and dimensions, some types of objects were counted by pairs or in groups of four. For example, in Fig. 12.5b, counting by pairs creates sixty-one counters worth two-ten-unit each in the first round, and six counters worth two-hundred-unit each in the second round. Similarly, in Fig. 12.5c, counting by fours creates thirty counters worth four-ten-unit each in the first round, and three counters worth four-hundred-unit each in the second round. Counting with multiple units further reduced the physical movements needed for counting, a significant gain of efficiency. Counting was still decimal – it proceeded from one to ten – but the amounts were understood as doubled for pairs (for example, two-hundred-unit, the double of hundred-unit) and as quadrupled for fours. It is not known whether conversion involved conscious manipulation, like we might mentally convert two dozen to twentyfour or two six-packs to twelve, or were unconsciously generated, by means of the lexical rules discussed in Chapter 5. The counting method influenced Polynesian numbers in several interesting ways: Counting by sorting created positional structure that extended into the tens of thousands and even the millions.27

27

Hale, 1846, p. 247.

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      Counting with multiple units shifted the numerical values of verbal terms upward by factors of two or four, especially in the peripheral regions of Polynesia. Number-words also changed morphologically to encode the type of item being counted, a practice known as objectspecified counting. For example, in the Tongan language, tefuhi and tefua, both words for ten-score, differentiated whether yams or coconuts were meant.28 Specifying the object implied the counting process that would be used with it, so that number-words were understood as reflecting counting by singles, pairs, or fours, and the same number-words took on different numerical values according to their contexts of use. For example, in Mangareva, rau meant hundred with an item counted singly but two hundred with an item counted by pairs.29 Such object-specified counting is found in many places around the world – not just Polynesia, but Mexico,30 South America,31 and Mesopotamia32 – but only rarely has it been associated with counting strategies. In some cases, number-words encoded the counting method directly. For example, the Hawaiian word for twenty, iwakalua, means nine and two, but why and how nine and two would mean twenty is not easily understood outside the context of counting pairs by sorting them: Nine pairs are set aside as counted, and the tenth pair, two items, is used as a counter.33 Two other variants of the counting method are worth describing. The first is a binary variant found in Mangareva (Fig. 12.6),34 the largest of the Gambier Islands that lie toward Polynesia’s eastern periphery. While binary counting begins in the same way that tallycounting does, by setting aside every tenth item, it then diverges in a way that capitalizes on the binary relations between one (counting singles), two (counting pairs), four (counting groups of four), and eight (counting by eights, a practice used with octopuses in Mangareva35)

28 31 34

29 30 Bender & Beller, 2007, Table 2, p. 220. Hiroa, 1938. Gallatin, 1845. 32 33 Bellamy, 2018. Overmann, 2021b. Overmann, 2020c, 2021c. 35 Bender & Beller, 2014, 2017. Laval, 1938, p. 211.

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    

. . Mangarevan binary counting for 741 items by (a) singles, (b) pairs, and (c) fours. As was true of tally-counting (Fig. 12.5), counting is decimal (one through ten), but the unit of counting (tauga) varies from one to four items. The first round sorts every tenth item, just as in tallycounting; thereafter, objects are counted by eights (varu), fours (tataua), twos (paua), and units (takau). Image by the author.

and the associated concepts and terms in language: eights (varu, meaning eight tens in binary counting, but also the word for eight), fours (tataua, meaning four tens), twos (paua, meaning two tens), and units (takau, meaning ten). Like tally-counting, binary counting can be used with singles, pairs, and fours. Unlike tally-counting, remainders are more easily accounted for. Tally-counting tends to round off all but the most significant digit produced by the final round of counting and subsume any remainders produced by previous rounds as a “remainder.” In the binary variant, after the first (decimal) round, all of the digits are produced by dividing the counters by eight, four, two, and one, and no conversions between units are required.

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      In Papua New Guinea, a similar strategy is used to count yams in groups of six.36 In these senary systems, exponential structure can reach an impressive seventh power of six, rivaling that of Polynesia, where counting generally reached the sixth power of ten (million) before significant European influence had occurred.37 The process of physically moving yams in order to count them differed in several respects from the sorting method used in Polynesia: The taitu [small yam] having been conveniently stacked, two men begin to tell them over. Each picks up three at a time, and they move off a few paces, and deposit them together. Meanwhile one of them, who acts throughout as teller, is shouting Nyambi, nyambi, nyambi, . . . (i.e. ‘One, one, one, . . .’). This means that they have put down the first unit of six. Without pausing they again take three each and as soon as these are deposited the teller changes his shout to Yenta, yenta, yenta, . . . (i.e. ‘Two, two, two, . . .’). So they proceed until six units of six have been deposited, when the teller throws one of his last handful to a third man sitting by, who places it before him as a counter to show that thirty-six taitu, or one peta, have been set down. The two men, however, do not pause, but count another six sixes, depositing them on top of the first peta; and, as they complete this second peta, the man who sits by silently places the second counter. So they go on until they have finished six peta, when they pause and the counters are carefully told over to verify. Five of these are thrown on to the heap, while one is kept as a major counter. By now there is a heap of 6  36 = 216 (less the one kept as a major counter) and this heap constitutes one storage heap called tarumba. It will form one separate heap on the floor of the mongo [storehouse]. But there still remain a great number of taitu, and the two men begin once more, depositing the tubers as they count them in a different place, and continue until they have accumulated a second tarumba. The process goes on almost without intermission, the teller repeating his count ‘One, one, one . . ., two, two, two . . ., three, three, three . . .,’ &c., up to ‘six, six, six, . . .’, when he begins afresh. When six tarumba have been deposited in six several heaps, a halt is called amid expressions of general satisfaction; one dameno has now been reached, i.e. 6 tarumba, or 6  216 = 1,296. (Williams, 1936, p. 226)

36

Williams, 1936; Evans, 2009; Döhler, 2018.

37

Hale, 1846, p. 247.

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     Beyond the physical and mental considerations involved in counting physical objects and their effects on counting behaviors, material forms, and numerical organization and structure, there is another potential influence on numerical language. This emerges from the circumstance that in Polynesia, forms of tally-counting were ceremonial and thus conducted publicly. This meant that the larger population was exposed to sequential recitation with truncated forms (e.g., one, two, three in the context of four hundred and eighty), a condition with the potential to lexicalize numberwords. As mentioned previously, lexicalization requires frequent usage, something predicted to take less time when large numbers – like those found in the amounts associated with tally-counting, which often reached into the thousands – are used with significant frequency. As enabled by the counting method and the ephemeral abacus it instantiated, Polynesian numbers accumulated and grouped. The counting sequences were also extensively related by heuristics that equated, for example, quantities like twenty singles, ten pairs, and five fours realized by changing the unit of counting from singles to pairs to fours. Yet an abacus made of coconuts or other commodities as they are being counted is not easily manipulated in the way needed for calculations more complex than simple arithmetic. Neither does it display results in way that is visually concise and persistent. Because of their weight and size, coconuts simply lack the potential for manipulation and concise display inherent in forms like the maize kernels and yupana used by the Inka. Both manipulation and display are essential to the apprehension of numerical relations and patterns, so the ephemeral abacus is not well suited for facilitating new conceptualizations of numbers, such as would be needed to elaborate numbers as a mathematical system. The ephemeral abacus also collapses the distinction between the objects being counted and the counting device. This is important because numbers are more easily conceived as entities distinct from what they are counting, the classic (if outdated) view

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      of numbers as “abstract,”38 when the material form used to represent them is something other than what they happen to be counting (this is discussed further in Chapter 14). Western researchers have not readily understood number systems based on counting processes, particularly those involving multiple units and object-specified counting. For example, counting with pairs has been misunderstood as comprising vigesimal numbers, rather than decimal numbers,39 and tally-counting as counting by elevens, rather than tens.40 Further, without the unifying structure provided by the underlying counting process, the different sequences in object-specified counting have been viewed as disconnected, mixed-base systems.41 Part of the problem in understanding such number systems is the “backward appropriation“42 that superimposes our Western ideas about what numbers are onto all numbers, regardless of their time or place. We also tend to count things one by one, and when we do count things by twos or fives, we change the counting sequence – two, four, six, eight, . . . or five, ten, fifteen, twenty, . . . – rather than keeping the counting sequence one, two, three, four . . .. Part of the problem is also the entrenched view that numbers are mental constructs that only later and under unknown circumstances become externalized materially, the neurocentric view of numerical origins. And with the ephemeral abacus, we also have a material form for counting that is not traditionally material in the way a wooden tally or counting board are. Nevertheless, the ephemeral abacus can explain the organization and structure of both conceptual and verbal forms of numbers across Polynesia, if we broaden our definition of what counts as material and are willing to admit material forms into the cognitive system for numbers. The historical invisibility of such

38 39

40

41

Overmann, 2018b. Kendall, 1815; Von Chamisso, 1821; more recently, see Lemaitre, 2004; Nishimoto, 2015. Von Chamisso, 1821, 1825; Balbi, 1826; Von Humboldt, 1839; Williams, 1844; Conant, 1896. 42 E.g., Lemaître, 1985; Bender & Beller, 2006. Rotman, 2000, p. 40.

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     forms and behaviors is not just a problem for archaeologists who research numerical origins; it is also a problem for linguists and psychologists, who find it difficult to explain how and why such numerical forms, both linguistic and conceptual, emerge from pure mentation. But given a material substrate like tally-counting and the associated insights from the challenges inherent in counting physical objects, everything soon falls into place: The linguistic and conceptual forms are based, once again, on material practices. Now that we have looked at two decimal, linear forms of abacus, we will consider one that is neither decimal nor linear: the 20  20 grid of the Yoruba cowrie counting system in West Africa.

    Cowries are sea mollusks whose shells were once used as currency throughout much of Asia and Africa. There are more than two hundred species of cowries, and they range in size from half an inch to several inches. Abundant in the Indian Ocean, cowrie shells were introduced into West Africa in minute quantities through an early overland route and in great numbers by Western nations trading cowries for slaves.43 Cowries have been used by West African peoples like the Yoruba and Igbo for centuries, and national accounts estimate their use as spanning thousands of years.44 Twenty thousand cowries, the amount of the largest bundle used in West Africa, weighed about what a man could carry.45 While each of the several number systems counting with cowries are interesting in their own right, here we will focus on the Yoruba number system because it has been reported to contain an abacus comprised of a twenty-by-twenty grid of cowries.46 The Yoruba cultural area today spans parts of Nigeria, Benin, and Togo centered above the Gulf of Guinea. The traditional number system of the Yoruba has been called “one of the most peculiar in

43 45 46

44 Hogendorn & Johnson, 1986. Mann, 1887; Johnson, 1921. Mann, 1886; Hogendorn & Johnson, 1986; Yang, 2019. Huylebrouck, 2006, 2019.

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     

. .. Yoruba numbers and the cowrie bundles they are based on. Data from multiple sources. Image by the author.

existence,”47 perhaps because numbers are conceived as arrays of combinatorial possibilities, rather than linear progressions. Yoruba number-words are formed, overtly and unambiguously, on bundles of cowries. While often called vigesimal,48 the Yoruba number system is not based on multiples of twenty as a vigesimal system definitionally would be (Fig. 12.7). For example, while there is a term for twenty squared (irínwó), there is not one for twenty cubed. Above two hundred, multiples of ten, rather than twenty, are productive: ten twenties (10  20, or 200), igba; ten two hundreds (10  200, or 2000), eɡbèwa; and ten two thousands (10  2000, or 20,000), òké. These _ _ _ _ amounts conform to different bundles of cowries: Five strings of forty cowries made a smaller bundle of two hundred (igba); ten smaller bundles made a larger bundle of two thousand (eɡbèwa); and ten larger _ _ bundles made a bag of twenty thousand (òké), the last being “a _ _ reasonable load for a man [to carry] over long distances” at about 23 to 46 kilograms (50–100 pounds).49

47 48

49

Conant, 1896, p. 70. E.g., Zaslavsky, 1970; Ẹkundayo, 1977; Verran, 2000b; Akinadé & Ọdéjobí, 2014; _ _ _ Babarinde, 2014. Hogendorn & Johnson, 1986, p. 76; also see Mann, 1886; Yang, 2019.

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     Yoruba numbers also do not appear to be overly based on digittallying, as the numbers between one and twenty are not divided into four groups of five, as is common in vigesimal systems because of the use of the hands and feet. Instead, the numbers eleven through fourteen are formed by adding the appropriate number to ten, while the numbers fifteen through nineteen are formed by subtracting the appropriate number from twenty. Subtraction is not unusual, as it is found in Roman numbers, where the words for eighteen and nineteen meant two from twenty and one from twenty. Sumerian also had subtractive forms, as 18 can appear as 20 minus 2 in Old Babylonian mathematical texts (as for example, the one shown in Fig. 13.9).50 Yet Yoruba is more subtractive than is typical, not just in forming fifteen as five from twenty, but also in forming the odd decades, hundreds, and thousands by subtracting, respectively, ten, hundred, and thousand from the next higher amount.51 The formations between eleven and nineteen suggest an influence of counting cowries by taking up five per hand to form groups of ten,52 as it would be easier to remove some from a higher group (twenty) than always add more to a lower group (ten). The formations for decades, hundreds, and thousands similarly suggest the use of cowrie bundles, adjusting them with half-bundles.53 Yoruba subtraction emphasizes two things: First, numerical organization and structure are influenced by the material forms used and how they are used, and second, these are matters driven by practical choices, not whim. Yoruba numbers preserve these material forms and choices as the cowrie bundling amounts and the use of both additive and subtractive forms. The really interesting thing about the Yoruba number system is not its subtractive forms, or even its overt basis in cowrie bundles, since we predict all numerical systems to have a material basis, but rather, its noncanonical expression of higher numbers.54 That is, there

50 51 53

Friberg, 2009; Friberg & Al-Rawi, 2016. Crowther, 1852; Armstrong, 1962; Ẹkundayo, 1977. _ 54 Johnson, 1970. Ẹkundayo, 1977. _

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52

Mann, 1887.

      is not just a single right way to form a number, like there typically is in the Western tradition. Noncanonical expression is not unusual in West African number systems associated with cowrie counting. For example, as was noted for the Igbo, another Nigerian ethnic group known for counting with cowries,

For it may be found that there is no set nomenclature and that though no mistakes are made in the actual counting, yet the native wording of the totals may vary widely. This difficulty can be demonstrated in English. Thus, the term 600 has a definite name in English but a native may express the same total as 50 in twelve places or as 60 in ten places: or as 100 in six places, or as 20 in thirty places. He will state that all are understood and no one method is more accurate or better than another. (Jeffreys, 1938, pp. 224–225)

In Yoruba, numbers can be expressed in a well-formed manner in multiple ways. For example, there are at least ten different ways to express 19,669, as shown in Table 12.1. This contrasts with the way a Western person would typically express the number: nineteen thousand, six hundred and sixty-nine, a formation that adds the values multiplied by the powers of ten in order from highest to lowest. The linguistic anthropologist Stephen Chrisomalis has noted that Western number formation is not absolutely inflexible in this regard, illustrating his point with 1,200,000, which can be expressed in English as one point two million, one million two hundred thousand, twelve hundred thousand, or even one two zero zero zero zero zero.55 Nonetheless, all these formations involve the exchange of decimal powers across a linear scale, something quite distinct from the fluid combinations of different multiplicative, additive, and subtractive formations seen in Yoruba. We would not typically express 1.2 million multiplicatively: two six hundred thousands, three four hundred thousands, five two hundred and forty thousands, six two hundred 55

Chrisomalis, 2020.

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     Table 12.1 Ten ways to form the number 19,669 in Yoruba. Data from Ẹkundayo (1977) and Akinadé and Ọdéjobí (2014); also see Crowther _ _ _ (1852) and Burton (1863). Yoruba expression

Numerical gloss

oke kan ó dín odúnrún ó dín _ _ _ mokanlélogbon _ _ _ oke kan ó dín irínwó ó lé _ _ mokandínlaadorin _ _ oke kan ó dín ojìlélodúnrún ó _ _ _ lé mesán-án _ eedegbààwá ó lé ójìlélegbeta ó __ _ _ _ lé mokandínlogbon _ _ _ eedegbààwá ó lé otalélegbeta ó __ _ _ _ _ lé mesán _ eedegbààwá ó lé orinélegbeta __ _ _ _ _ dín mokanlàá _ eedegbààwá ó lé eedegberin ó __ _ __ _ _ dín mokanlélogbon _ _ _ eedegbààwá ó lé egbeta ó lé __ _ _ _ mokandínlaadorin _ _ eedégbokandínlogorún ó dín __ _ _ _ _ mokanlélogbon _ _ _ egbèjìdínlogorún ó lé _ _ _ mokandínlaadorin _ _

(20,000  1) – 300 – (1 + 30) (20,000  1) – 400 + (1 from (10 from (20  4))) (20,000  1) – ((20  2) + 300) + 9 (1000 from (2000  10)) + ((20  2) + (200  3)) + (1 from 30) (1000 from (2000  10)) + ((20  3) + (200  3)) + 9 (1000 from (2000  10)) + ((20  4) + (200  3)) – (1 + 10) (1000 from (2000  10)) + (100 from (200  4)) – (1 + 30) (1000 from (2000  10)) + (200  3) + (1 from (10 from (20  4))) (100 from (200  (1 from (20  5)))) – (1 + 30) (200  (2 from (20  5))) + (1 from (10 from (20  4)))

thousands, ten one hundred and twenty thousands, or one hundred and twenty ten thousands. We would not typically express 1.2 million by subtracting from two million: one eight hundred thousand from two million, twice four hundred thousand from two million, or two hundred thousand from one million from two million. This is difficult enough, even though 1.2 million is a fairly simple number; the difficulty of generating alternative expressions increases with the level of detail a number has. Our Western numbers do not train us to think about or manipulate numerical relations in ways that would let us express a large and detailed number like 19,669 in two or three

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      multiplicative ways, let alone ten, let alone with a mix of subtractive and additive forms, let alone without writing. It is a very different conceptualization of number, one that is not just combinatorial but implies an impressive mental faculty in manipulating and keeping track of the different combinations, relations, operations, and remainders. How quickly an individual with an average education and knowledge of mathematics might be able to articulate multiple ways of forming any particular number in Yoruba is currently unknown. The fact that the ten alternatives in Table 12.1 are split between two different publications written nearly forty years apart suggests that some of them might be fairly difficult to generate, even if all of them are ultimately well formed. Unfortunately, the rapid and ongoing displacement of the traditional Yoruba system by Western numbers and/or Yoruba numbers revised as a decimal system56 means that generating alternative names is likely to become increasingly difficult as proficiency with the traditional number system is lost; this also means that even if the ability were measured today, the result is unlikely to represent past levels of proficiency with any degree of accuracy. Forming number-words combinatorially, rather than linearly, suggests that Yoruba numbers (and other West African number systems that counted with cowries) may have been realized and elaborated with a manipulable technology like cowries or pebbles, with fixed forms of notation like tallies perhaps following later. It should be noted that keeping records with manipulable rather than fixed forms is a well-known practice, both in Africa and in the Near East. For example, the Kingdom of Dahomey, a West African state that flourished between the seventeenth and nineteenth centuries in what is now Benin, took periodic census of its population by collecting pebbles, one from each person; these were placed into sacks that were labeled to differentiate demographic characteristics like gender and 56

Adeyinka, 2010.

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     age and then stored to represent the count.57 The numbers involved were not inconsequential: The Dahomey population was estimated to consist of several hundreds of thousands.58 The numerical tokens of Mesopotamia also fall into the category of records kept in a manipulable form, albeit with the elaboration that shape and size were used to keep track of commodity information, rather than labeled containers. The earliest impressions (made with tokens) and the notational systems that developed subsequently in Mesopotamia were also linear in both their physical organization and the ways in which they were combined, implying that tokens likely developed subsequent to linear forms like the tally, which is consistent both with archaeological finds of bones with linear striations in the Palaeolithic Levant59 (note the caveats in Chapter 11), textual records from the early second millennium BCE describing the use of tallies,60 and unambiguous tallies in the first millennium BCE.61 The fluid combinatorics used to form Yoruba numbers evokes the marks on the Ishango tally (Fig. 11.6), which combine an impressive insight into addition, multiplication, and the indivisibility of prime numbers – matters that would depend on a manipulable technology being used – with notational clumsiness. The combination suggests a manipulable technology as an early form with a pervasive and persistent influence on numerical organization and structure. If the Ishango artifact and the numerical tradition it represents were ancestral to West African cowrie-counting systems, related in the same way that Western numbers still reflect some aspects of Babylonian numbers across a span of thousands of years of time, it would demonstrate an impressive depth of time for African numeracy at more than 20,000 years (Chapter 11). Combinatorial fluidity in forming number-words is also consistent with the twenty-by-twenty cowrie grid said to have been used by

57 59 61

58 Herskovits, 1932. Herskovits, 1938. E.g., Tixier, 1974; Reese, 2002; Prévost et al., 2021. Henkelman & Folmer, 2016.

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60

ETCSL, 2005.

     

. . Twenty-by-twenty cowrie grid showing the final result of multiplying nineteen by seventeen. Image by the author based on material in Huylebrouck (2006, Fig. 22 right, p. 156, and 2019, Figs. 6.5, 6.6, and 6.7, pp. 111–113).

Yoruba for calculation (Fig. 12.8). The grid has been described in two publications, both written by the Belgian mathematician Dirk Huylebrouck.62 Huylebrouck does not say where he encountered the practice, in 2006 incorrectly attributing it to the 1887 article by Adolphus Mann, and more recently demurring to elaborate beyond his published work.63 Despite the uncertainty posed by the unknown provenience and single-author source, the grid cannot be dismissed

62 63

Huylebrouck, 2006, 2019. Personal communication from Dirk Huylebrouck dated November 28, 2021.

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     out of hand. This is because it produces answers that conform to Yoruba spoken numbers, which work on a combinatorial logic quite different from that of the linear counting board and abacus. As an example of multiplication, Huylebrouck stipulates that four hundred cowrie shells are first arranged in twenty groups of twenty, each of which consists of four rows of five. Then, to multiply nineteen by seventeen, the procedure is as follows: From each group of twenty, one cowrie is set aside, one being the difference of the base, twenty, and the multiplicand, nineteen. The set-aside shells are reformed as an individual group of twenty. Of the twenty groups now with nineteen shells each, three groups are singled out, three being the difference of the base and the multiplier, seventeen. The setaside shells are rearranged as two groups of twenty and a remainder, seventeen. The total shells removed from the original grid are now three groups of twenty and one group of seventeen. The product of nineteen by seventeen is four hundred (the original grid) minus what has been set aside, sixty (the three groups of twenty) and seventeen (the remainder), amounts that correspond to the typical Yoruba verbal expression: seventeen from sixty from four hundred, which is equal to what a Western person would call three hundred and twenty-three. Experimentation with the grid and associated formula (x2 – (x) (x – a) – (x – b)(a), where x is the base number, a is the multiplicand, b is the multiplier, and the whole reduces to a  b) shows that any number divisible by four can act as the base, including four. That is, the base need not be a square number, but it must be divisible by four so that the shells form a rectangular layout. As a matter of functionality, the grid should be big enough in size to accommodate the multiplication of large numbers, but not so big that the amount of shells becomes unmanageable. For example, using 24 as the base creates a grid of 576 shells, almost half again as large as 400, when 400 shells are already a lot of moving parts to set up and keep track of. Both multiplicand and multiplier should also be fairly large (i.e., close to the selected base), since the smaller these numbers are, the more shells must be moved to perform the calculation (e.g., multiplying

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      Table 12.2 Possible variations of movements on the Yoruba cowrie grid. The final variant is the most efficient, as it requires the fewest movements; the first variant, the one described by Huylebrouck, is the second most efficient. Algorithm

First step

Second step

Third step

Singles first (19  17) Singles first (17  19) Groups first (19  17) Groups first (17  19)

Move 20 single shells Move 60 single shells Move 1 group of 20

Move 3 groups of 19

Rearrange

Move 1 group of 17

Rearrange

Move 57 single shells Move 17 single shells

Rearrange

Move 3 groups of 20

Not needed

even 15  15 requires moving nearly half the shells). As smaller numbers are usually easier to multiply mentally, using a grid would also constitute unnecessary effort. There are, in fact, four ways of moving the shells to achieve the same result (Table 12.2); some of these algorithms are more efficient in terms of the number of movements involved and whether the shells must also be rearranged to form a coherent answer. Interestingly, the first algorithm (single shells removed first, 19  17), which is the one that Huylebrouck described, is not the most efficient of the four, as we might predict for a practice implied to have significant longevity. The second algorithm exchanges the terms and is the least efficient in terms of physical movement, showing that when single shells are removed first, the larger term should also be first (i.e., the term with the smaller difference from the base number of 20, in this case, 19, as in the first algorithm). The next two algorithms remove groups before single shells; the third algorithm is inefficient. The fourth algorithm (groups of shells removed first, 17  19) requires moving the fewest shells and does not require any subsequent rearranging; thus, the most efficient algorithm is removing groups first with the smaller term first

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     (i.e., the term with the larger difference from the base number of 20, in this case, 17). Let us say the grid was determined to be a relatively recent invention, rather than a cultural practice enduring for centuries or millennia. It is still significant that it permits so many different ways of moving to perform the same calculation, something that linear forms like counting boards and abacuses generally do not. And the flexible combinatorics of Yoruba numbers and other cowrie-based counting systems like the Igbo system still confound the linear logic of decimal progression and the successor function. They contradict the assumptions that all numbers are just like Western numbers and that numbers are mental entities only later stamped onto passive material forms. Yoruba and other cowrie-based numbers are not difficult to explain if the relation is simply reversed to have the concepts follow from the material forms and practices. In any case, their overt material basis suggests that this is the correct way to view them, and in this they are not exceptional, as cross-cultural comparisons of number concepts and material forms and practices suggest that the former follow the latter as a general rule. All the number systems reviewed in this chapter accumulate and group. These number systems also show, albeit in distinct ways, that different material forms – maize kernels, coconuts, and cowries – have different potentials for manipulating numerical information for the purpose of calculation. Counting by sorting coconuts, for example, is less adaptable to the kind of manipulation needed for calculating because coconuts are big and heavy; counting with cowries is more adaptable in this regard because of their smaller size. These differing potentials emerge from the physical forms and substances used, and how their properties of weight, size, and viewability interact with human behaviors and psychological capacities. None of these manipulable forms produces a record of numerical information, which can motivate the use of a fixed device of some sort for this purpose. In the case of the Inka, that device was the khipu; for many other societies, the device is the handwritten notation, which is examined in the next chapter.

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 Handwritten Notations

Like their predecessors, handwritten numerical notations develop from, and thus reflect, the capabilities and properties of the technologies that preceded them, things like fingers, tallies, and tokens (Table 7.1). Thus, notations both accumulate and group, not because of some kind of innate predisposition for a concept of number with these qualities, but rather, because the material devices that preceded them accumulated and grouped. Like each of their precursors did, notations also respond to the limitations of their predecessors, for example, by providing the persistence in recording that manipulable forms cannot. Notations also bring new capabilities and limitations to the cognitive system for numbers, for example, adding conciseness and being fixed rather than manipulable. Their conciseness lets notations represent numbers at an unprecedented volume, enabling the compilation of tables of relations that influence numbers toward being conceived of in terms of their relations; their fixedness motivates the development of algorithms based on the knowledge of numerical relations, rather than the physical movements of elements like beads on an abacus. In sum, notations are part of the chronology of material forms for numbers, albeit the last to emerge and most elaborated form known. As such, numerical notations share a continuity of descent with precursors like tokens, tallies, and fingers. Numerical notations also share contiguity of function with their precursors: Material forms like notations and fingers are meaningful as numbers because they instantiate quantity. That is, they mean quantity by being quantity, achieved through repetition. For example, three tokens or proto-cuneiform cones, three cuneiform wedges, the three vertical strokes in the Roman numeral three, and the three horizontal strokes in the Chinese numeral three all mean 

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    

. . Instantiation: to represent by being an instance of something, in this case, quantity. Numerical notations mean three by being three in the quantity of their elements: (a) three fingers; (b) three tally notches; (c) three knots in a quipu; (d) three abacus beads; (e) three small cones used in Mesopotamian tokens, impressions, and proto-cuneiform notations; (f) three cuneiform wedges; (g) Roman numeral three; (h) Chinese numeral three. The Western numeral “3” is a cursive version of (f) and (g). Images are in the public domain.

three in virtue of having three elements, in exactly the same way that three fingers, three tally notches, and three abacus beads mean three (Fig. 13.1). All these material forms share function (how they mean). This means that their morphology (what they look like) can be used to represent other information, like bundled value (e.g., as the shapes and sizes of the Mesopotamian tokens were used to encode different values or commodities). The continuity of descent and contiguity of function that connect

written

numerals

to

their

precursors

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are

generally

   underappreciated, for three reasons. First, Western numerals are “ciphered”: They are conventions whose meanings must be learned because they are not directly expressed by the forms themselves. For example, “7” is a sign that does not consist of countable elements like those shown in Fig. 13.1. However, “7” descended from written forms with seven elements (e.g.,

).1 The Mesopotamian cuneiform

numerals joined with those from Egypt, Greece, and India to produce the familiar Western numerals 0 through 9. The resultant ciphered forms can make it difficult for us to appreciate the origin of Western numerals as signs that meant number by instantiating it, the point at which numerical notations most clearly display their continuity and contiguity with precursors like tokens, tallies, and fingers (as can be clearly seen in Fig. 13.1). The second reason we fail to appreciate the continuity and contiguity shared by numerical notations and their precursors is that numerical notations are usually analyzed as a standalone technology, as if they emerged without any predecessors to realize the qualities they embody and represent. This is likely because few of the technologies appear in unambiguous chronological relations; the Mesopotamian notations and their predecessors are unusual in this regard. And the third reason is that the differences between notations and the other material devices used for representing and manipulating numbers are often compelling. We tend to view numerical notations as symbolic, immaterial, and ephemeral, qualities that solidly physical devices like tallies and abacuses seem to lack. However, as previously mentioned, here we are taking a different approach, seeing numerical notations as a material form, albeit one that possesses additional qualities that their precursors either lack or do not have to the same degree, like conciseness. In this chapter, we will examine four issues. The first concerns the ways in which written numbers and their unwritten precursors share how they mean, while the second concerns the ways in which 1

Chrisomalis, 2010.

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     written numbers differ in the way they mean from notations for nonnumerical language. These two issues are related: Written numbers differ from nonnumerical notations because, like their precursors, they instantiate quantity, or represent by being an instance of what they mean (until they become ciphered, that is). In contrast, nonnumerical notations signify, or represent by depicting, indicating, or suggesting their meaning through resemblance and convention. The third issue we will examine in this chapter concerns the unique capabilities, properties, and functions that written numbers and writing in general add to the cognitive system for numbers. For example, written numbers are highly concise, which makes them able to record and simultaneously display larger volumes of numerical information than their predecessors could; they are also handwritten, which means that they involve the distinctive neurological and behavioral reorganizations associated with literacy. As we will see, these attributes influence numbers to be reconceptualized as entities in a relational system, where their meaning derives more from the patterns they make than from any individual number alone, similar to the way notes combine to make music and sounds combine to form language.2 The fourth and final issue concerns the major trends that emerge when the material forms used for numbers, including handwritten notations, are chronologically sequenced. One trend is the increase in an implicit component that consists of the knowledge the user must supply in order to use a particular material form, things like place value that the material form does not explicitly represent.

         To look at how numerical notations and their unwritten precursors share the function through which they are meaningful, let us compare the impressions that emerged in Mesopotamia around the fourth millennium BCE to the clay tokens used as numerical counters 2

Plato, 1892.

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  

. . Correspondences of shape and size (top, plain tokens used for accounting; middle, impressions and proto-cuneiform notations; bottom, cuneiform numerals). Only the system for counting most discrete objects is shown for simplicity and ease of comparison. Tokens adapted from Englund (2006, Fig. 24, p. 29); proto-cuneiform notations from Nissen et al. (1993, Fig. 28, pp. 28–29); the cuneiform numerals are Unicode characters. Image by the author. Previously published in Overmann (2019b, Fig. 9.1, p. 158).

(Fig. 13.2, top). The generally accepted account of the invention of Mesopotamian writing has numerical impressions emerging to display the accounting tokens contained inside bullae, clay spheres used to enclose the tokens; this innovation likely responded to the need to know what was inside a bulla without having to crack it open and thus destroy its integrity as a container, for once it was sealed, there was no way to know what was actually inside. At some point, the redundancy of tokens and impressions was resolved by using impressions made on clay tablets for recording, while tokens remained the manipulable technology suitable for calculating.

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     Both tokens and impressions encoded numerical values through repetition (element quantity) and conventions of shape and size (element value). Here we want to focus on the morphological correspondences between the two. In general, these are important because they tell us that the tokens had numerical meaning; the numerical meaning of the impressions is then anchored by their correspondences with the later proto-cuneiform notations.3 But our focus is even more basic: The tokens and impressions had the same shapes, sizes, and quantities; in some cases, the impressions even appear to have been made with the tokens themselves. This morphological correspondence means that tokens and impressions had the identical function: Both represented numerical meaning by instantiating it. Three tokens and three token-shaped impressions meant three by virtue of having three elements. They differed in that tokens were convex, manipulable, and produced by molding clay, while impressions were concave, fixed, and produced by impressing clay with objects of the requisite shape and size. This difference between manipulability and fixedness made tokens suitable for manipulating (calculating) and impressions suitable for representing (recording) numerical information. Tokens (Fig. 13.2, top) and impressions were preceded by fingers and tallies, forms that are attested textually and archaeologically. These too represent numerical meaning by instantiating it through repetition (Fig. 13.1), with the limitation that their morphological forms are less easily adapted to express things like value and commodity. Tokens and impressions were followed by protocuneiform notations (Fig. 13.2, middle), which were followed in turn by cuneiform numbers (Fig. 13.2, bottom). Both were marks on flat surfaces made by means of a reed stylus, a form of handwriting. Proto-cuneiform notations retained the conventions of shape and size used to encode value and commodity in the impressions and

3

Broman, 1958; Oppenheim, 1959; Amiet, 1966, 1972, 1987; Schmandt-Besserat, 1992a; Nissen et al., 1993.

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   tokens. In the cuneiform numerals, the conventions representing values were simplified and reduced, while the conventions representing commodities were discarded. Commodities were instead labeled with small pictures and conventions that had emerged in conjunction with proto-cuneiform. By the time cuneiform numerals emerged, these small pictures and conventions were fast becoming a writing system capable of expressing nonnumerical language with relative fidelity. Importantly, handwriting notations would realize neurological and behavioral/biomechanical effects that would influence the form of written characters, both numerical and nonnumerical; how these changes would affect the subsequent elaboration of numbers as a mathematical system will be examined in the next sections. When we recognize numerical notations as sharing continuity of descent and contiguity of function – instantiation – with unwritten precursors like tallies and tokens, we gain new insight into threedimensional forms of recording, things like the Mesopotamian tokens, Inka khipu, and Pomo sticks. Such technologies tend to be viewed as quite different from written notations, and perhaps a bit mysterious, which is an interesting phenomenon in and of itself, since three-dimensional forms have a much longer history and prehistory of being used for recording and calculating than written forms do.4 Contemporary researchers are still trying to understand what the tokens used in Neolithic accounting might have meant for Mesopotamian numeracy and recordkeeping.5 Similarly, while the Conquistadors were impressed by the khipu they encountered in Peru, they never did understand them very well;6 one early chronicler remarked that the use of knotted strings as writing simply had to be seen to be believed.7 Granted, the nonnumerical component of the

4 5

6 7

Adkins, 1956. MacGinnis et al., 2014; Bennison-Chapman, 2018; Overmann, 2018b, 2021b; Valério & Ferrara, 2022. Prescott, 1847; Andrien, 2001; Brokaw, 2010. Polo de Ondegardo, 1561; quoted and translated in Brokaw (2010, p. 130).

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     khipu remains undeciphered and untranslated to this day; it likely represented mnemonic conventions that were meaningful only to those who had learned them. The numerical component of the khipu, however, is readily accessible and overtly meaningful as numbers; this is because, like tokens and other notational and unwritten forms of numbers, it instantiates quantity. The khipu is worth examining in some detail to understand not just how it means through instantiation, but also what material form implies about elaborational potential, an area where handwritten signs are advantageous. The device itself (Fig. 13.3) was “a cord about two feet long, composed of different-coloured threads tightly twisted together, from which a quantity of smaller threads were suspended in the manner of a fringe. The threads . . . were tied into knots; the word quipu, indeed, signifies a knot.”8 Possible khipus have been dated to as early as the mid-third millennium BCE.9 By the time the Spanish arrived, some 4000 years later, khipus were an essential part of the recordkeeping and flow of information used to govern the Inka Empire,10 whose geographic extent included modern Peru, much of Chile, and portions of Bolivia and Ecuador. Various chroniclers described the use of knots and threads (or cords or strings) in recording and reporting numerical and statistical information. Khipus were used to collect demographic statistics (marriages, births, deaths, clans/ moieties, social ranks, occupations), as well as communicate data about tribute, revenue, commodities, and raw materials.11 Beyond recording numerical information, khipus were also said to have been used to record “personal inventories, censuses, laws, ritual sacrifices, religious geography, calendrical data, and, perhaps most intriguingly and certainly most controversially, narrative histories.”12 However, “no specific khipu has ever been reliably identified as a narrative text.”13 Today, khipus remain in limited use among Quechan

8 11 13

9 10 Prescott, 1847, p. 118. Mann, 2005; Turnbull, 2013. Brokaw, 2010. 12 Prescott, 1847; Medrano & Urton, 2018. Brokaw, 2010, p. 96. Hyland, 2017, p. 412.

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  

. . Inka khipu made of knotted cotton or wool, found in Pascasmayo Valley, Peru and dated to 1430–1530. The artifact contains 75 threads and measures 71  104 centimeters (28  41 inches). Image © Trustees of the British Museum and used with permission. Inset: Geographic extent of the Inka Empire at its height in the early sixteenth century. Adapted from an image in the public domain.

villagers,14 and similar forms are used for mnemonic purposes in the Amazon.15 Khipu encoded an unambiguously decimal (base 10) system of numbers.16 Knots of different types distinguished the units from the higher multiples of ten (e.g., tens, hundreds, thousands, and, very rarely, ten thousands17), while reinforcing the order in which knots were read (Fig. 13.4). For units, figure-eight knots (∞) represented the value one, while long knots with two (||) to nine (|||||||||) twists represented the values two through nine; for tens and higher values, single knots (●) were repeated up to nine times (●●●●●●●●●).18 Within each

14 16

15 Medrano & Urton, 2018. Chaumeil, 2005; Hugh-Jones, 2016. 17 18 Locke, 1923. Chrisomalis, 2010. Ascher & Ascher, 1981.

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    

. . Notional Inka khipu. Figure-eight (∞) and long (|||) knots were used in the units position (the loose ends of pendants and subsidiaries), while single knots (●) were used for higher multiples of ten (tens, hundreds, thousands). Adapted from Ascher and Ascher (1981, Fig. 2.7, p. 17). Image by the author; key after discussion in Ascher and Ascher (1981) and an online image by Gina Cronin.

cluster of knots, one to nine single knots (or the unit analogies in figure-eight or long knots) were added to represent the digit, and a space of untied thread where a cluster might otherwise appear was understood as not containing a number. Spaces between clusters distinguished both the units and the higher multiples of ten, with knots representing the units placed at the far (loose) end of a cord and knots representing tens and higher multiples of ten placed above them toward the end connected to the main cord. The khipu is often interpreted as a cumulative-positional number system, cumulative because the knots (or their turns) were repeated within each cluster with the expectation they would be added together, positional with the digits (knot clusters) each multiplied by the appropriate power of ten before being added to realize the

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   total value.19 Thus, a cord containing clusters of one, two, and three single knots and a long knot with three twists, read from the main cord to the loose end – ●, ●●,●●●, |||, as in the rightmost thread on the notional khipu in Fig. 13.4 – would represent the number 1233: (1  1000) + (2  100) + (3  10) + (3  1). However, since knots take their value from their physical position on the string, the identical knots also support an additive interpretation20 (Fig. 12.4). This more parsimonious explanation makes the khipu a cumulative-additive system, cumulative as given above, additive with the digits valued differently according to their spatial placement before being added to realize the total value. Read as a cumulative-additive number system, 1233 would be (1000) + (100 + 100) + (10 + 10 + 10) + (1 + 1 + 1). Here the encoding resembles that of Roman numerals, where the number 1233 would be MCCXXXIII. However, Roman numerals additionally incorporated a sub-base of five, as well as back-counting conventions (e.g., IV or one before five instead of IIII, XL or ten before fifty instead of XXXX, etc.) that helped mitigate the difficulty of appreciating quantities at the subitizing limit. It would be difficult indeed to represent things like sub-bases and back-counting with knots, let alone calculations and equations, since knots cannot be morphologically varied in the manner or degree possible with writing, which has a much greater capacity for visible alteration in this regard. The difficulty of appreciating more than about four knots might also explain why reading khipus has historically been described as involving touch and not just vision: Touch was needed to ascertain the quantity of knots, something that was more difficult to do by visual inspection alone. Relative to written notations, knots would not have been as adaptable to some of the purposes for which writing has elsewhere been employed in number systems, as described in the next sections. Like the difficulty of calculating with an abacus made of coconuts, the material form of knots might have effectively limited the potential for 19

Chrisomalis, 2010.

20

Florio, 2009.

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     further elaboration as a mathematical system. Written notations, in comparison, are not limited in these same respects.

         Probably the most interesting thing about handwritten notations is that they are not generally considered to be a material form. Instead, we tend to think of the things we write with as the material forms involved in writing: pens, pencils, chalk, ink, and chisels; paper and chalkboards and stone; typewriters and computer keyboards. In comparison, we think of what we write, the various combinations of characters forming words and sentences and numbers, as symbols that express our meaning. We also think of written characters as ephemeral because marks on surfaces are relatively impermanent as material forms go. From an archaeological view, writing can identify the culture associated with excavated artifacts, albeit one that is very recent, since writing systems are a post-Neolithic phenomenon. Beyond this diagnostic value, writing in an archaeological context is often more compelling for what it might say once it has been translated – for what it can tell us about ancient peoples in their own words – than for what it is in its own right as a material form. But in this regard, handwritten notations also admit opportunities to consider how things mean – how material signs are meaningful, as discussed in Chapter 8 – rather than just simply what they mean, which involves their translations or the implications drawn from where they were placed or found, or why they mean, which is the more intractable problem of meaningfulness. Another reason we tend not to think of written notations as a material form is that writing is often seen not as expressing language but as being language.21 While this does not answer the question of why writing is not considered material when it is a physical form that we interact with manuovisually, numerical notations are at least 21

Overmann, 2022.

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   understood as having a role in how we acquire numbers, what we understand them to be, and how we use them.22 The position that (all) writing (including numbers) is (just) language may be consistent with the view that numbers emerge from language,23 but it does not acknowledge, let alone explain, the highly visual and essentially alinguistic nature of numerical thinking, or why written and spoken forms of number differ so significantly. Spoken numbers, for example, are irregular in ways that have no counterpart in written forms. In English, the spoken numbers one through ten are atomic, eleven through nineteen are irregular, and twenty and higher are regular, while the written numerals used with English, whether Western, Roman, binary, hexadecimal, or other, are regular in all these ranges. Conversely, alphabetic numbers – notational systems in which the units 1–9, tens 10–90, hundreds 100–900, and so on are expressed with unique symbols – have no counterpart in any known natural language.24 Numerical notations share function with their unwritten precursors, as well as embody properties like accumulation and grouping realized by means of them, and this contiguity and continuity also lack counterparts in natural languages, which have no material form until writing is invented. Taken all together, these matters argue that written notations have far more in common with their material precursors than they do with spoken numbers. In original writing systems like the one that emerged in Mesopotamia, written notations can be divided into two basic types: numbers and nonnumerical language. Notations for numbers mean quantity by instantiating it: Three small cones in proto-cuneiform and the three vertical wedges in the cuneiform number three, like the three vertical strokes in the Roman numeral three and the three horizontal strokes in the Chinese numeral three, all mean three in virtue of having three elements (Fig. 13.1). They mean three because

22 24

Landy et al., 2014; Schlimm, 2018. Chrisomalis, 2010.

23

Overmann, 2022.

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    

. . Signification: to represent by depicting, indicating, or suggesting something: (a) Mesopotamian sign depicting cow; (b) Chinese sign depicting sheep; (c) Mesopotamian convention for sheep; (d) Chinese convention for field. Images are in the public domain.

they are three. In contrast, early nonnumerical signs signified their meaning

through

resemblance

and

convention.25

In

early

Mesopotamian and Chinese writing, forms that resembled included the pictures of cows and sheep that meant cow and sheep; conventions included the quartered circles that meant sheep or field (Fig. 13.5). These forms signify, rather than instantiate: They are not themselves cows, sheep, or fields in the same way that three of anything is three. The distinction between signification and instantiation was made in 1929 by the Belgian surrealist René Magritte when he painted a pipe accompanied by the words, “Ceci n’est pas une pipe” (“This is not a pipe”). That is, the painting, known as “La Trahison des Images,” is not itself a pipe, which would be instantiation; rather, it is an image of a pipe, which is signification. This fundamental difference – instantiation versus signification – is related to how the two types of signs mean as material forms. Numerical signs instantiate particular quantities by repeating their elements the requisite number of times. Recognizing the quantity of elements involves the perceptual system for quantity, and this is a major influence on the way these forms change over time (Fig. 13.6).

25

Overmann, 2016a, 2021a.

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  

. . Change in written numerals over five millennia and multiple languages. (Top) Small (subitizable) numbers change the least; the Western numeral “2” is a cursive form of two vertical strokes. (Bottom) In contrast, large (nonsubitizable) numbers change to a greater extent, becoming forms that avoid the need to count elements or element subgroups or (not shown) being replaced by conventional forms (like the X in Roman numerals means ten) understood as bundled values. Data from Chrisomalis (2004, 2010). Image by the author. Previously published in Overmann (2022, Fig. 7, p. 25).

When the elements represent a subitizable quantity, the sign will likely remain as a form with countable elements, perhaps conserving over thousands of years, because subitizable quantities are inherently appreciable (Fig. 13.6, top).26 In contrast, when the elements represent nonsubitizable quantities, they will either be replaced by conventions meaning bundled values that reflect material forms like the fingers (e.g., in groups of ten), or they will be rearranged into subitizable subgroups whose quantity is appreciable but requires counting (Fig. 13.6, bottom, Egyptian hieroglyphic). Over time, through the ways in which writing changes discussed below, signs consisting of subitizable subgroups will become forms that avoid the need to count elements or element subgroups. In the figure, this change can be seen in Egyptian hieratic, which is a cursive form of the hieroglyph to its left; in Egyptian demotic, the countable elements are no longer apparent, and the sign has become ciphered. As conventions and ciphers, numerical signs become subject to the processes that change written signs, described below. Even as conventions and ciphers, numbers

26

Overmann, 2021g.

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     remain uniquely identifiable as numbers because they embody numerical relations that have no counterpart in signs for nonnumerical language. In comparison, signs that signify, either by resembling an object or representing a convention, leverage our capacity for recognizing physical objects. A part of the temporal lobe, the fusiform gyrus, has an evolutionary function for recognizing objects and faces, and it becomes trained to recognize written characters as if they were physical objects.27 We recognize physical objects topologically, through combinations of their structural features and spatial relations. Topological recognition had a significant effect on ancient writing systems because, as writers started to recognize written characters through combinations of their structural features and spatial relations, the need for the pictures and conventions to resemble their ideal forms was relaxed.28 This meant that their forms could lose their depictiveness (or deform) and still be recognized (Fig. 13.7). Relaxing the need to resemble also freed the elements in the set of characters to converge on points of contrast that aided their discrimination, the ability to tell characters apart from one another, and individuation, the ability to identify characters as themselves. Points of contrast were the lines, orientations, and shapes that activated the visual system more strongly.29 And as characters became less depictive, increased amounts of training were required, because topological recognition meant training the fusiform gyrus to recognize character features and relations. Today, the process of acquiring the specific neurological reorganizations involved in training the fusiform gyrus is simply known as learning to read and write. Other influences changed the form of written characters. One had to do with tolerance for ambiguity, the ability to recognize characters topologically despite their deformation. This sacrificed conformance to an ideal, easily recognized form – handwriting became

27 28

McCandliss et al., 2003; Dehaene & Cohen, 2007; Coltheart, 2014. 29 Overmann, 2016a, 2021a. Hodgson, 2007, 2012, 2019.

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  

. . Loss of depictiveness in early writing. Archaic forms were meaningful because they resembled objects or had conventional significance (from left to right: fish, bird, axe, arrow, and vase). After a thousand or more years, written forms no longer resembled, an effect of topological recognition. Adapted from de Morgan (1905, Fig. 38, p. 243). A version will appear in Overmann (2023a).

sloppy – but it gained speed of production, enabling writing to become a tool that could keep up better with the speed of thought.30 Another influence on written form had to do with the amount of complexity: High levels of detail make characters easier to recognize, particularly

30

Overmann, 2016a, 2021a.

https://doi.org/10.1017/9781009361262.015 Published online by Cambridge University Press

     for novices, but they also take longer both to write and read. Highly detailed characters thus slow proficient readers, who are able to make greater use of contextual clues to meaning.31 Over time, written forms simplified, suggesting a balance in the amount of detail needed by proficient and novice readers. Other influences on form involved standardization and automaticity, the ability to produce characters the same way every time without thinking about it;32 this freed resources like attention and working memory to focus on the content of writing, rather than the mechanics of its production. There were also biomechanical influences on written forms, which reduced the amount of physical effort involved in writing;33 besides simplification, another result was cursive, a form of writing whose “abbreviated signs, crowded writing, and unclear sign boundaries”34 imply both tolerance for ambiguity in sign form and reduced physical movement across and above the writing surface. This fundamental difference in how signs for numbers and signs for

nonnumerical

language

represent

–

instantiation

versus

signification – has consequences for how the two sign types develop over time. Pictures and conventions do not specify the intended words (or morphemes). For example, in Mesopotamian writing, a picture of a head could mean head, person, or capital. This semantic range means that pictures and conventions are inherently ambiguous regarding the words they are intended to express, so they must be specified in some way if they are to be understood as specific words. This specificity is achieved by adding conventions that identify the type of word (i.e., determinatives) or by adding clues to their sound values (phonography). Numerical signs, in contrast, are not under any pressure to be specified as words. As signs with countable elements, they instantiate quantity appreciable through the (alinguistic) perceptual system for quantity, and as conventions and ciphered forms, they represent

31 33

Bird, 1998, 1999; Ravid & Haimowitz, 2006. 34 Zipf, 1949. Veldhuis, 2011, p. 72.

32

Taylor, 2011; Bramanti, 2015.

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   numbers understood through a cognitive system that is highly visual in nature.35 Thus, they do not need to add clues that specify the associated number-words in terms of their sound values, and in fact, adding such elements to numerical signs increases their visual complexity in a way that degrades or destroys their value as numbers. Note the difference between “7” and “seven”: While both mean seven, the former is concise in a way that facilitates the recognition of relations and patterns, while the latter is complex in ways that make it less recognizable and usable as a number. (Readers interested in exploring this difference might try using phonetic forms in calculations; they should find that phonetic forms are less likely to reveal patterns and relations, less convenient to use, and slow the process by taking longer to write.) Because written numerals are signs that are semantically meaningful but phonetically unspecified, the same numerals can be used with different languages. For example, the Western numerals (0 through 9) are used throughout the world, across a wide variety of languages and scripts; similarly, we use Roman numerals without knowing or needing to know their names in Latin. The form of numerical signs also changes over time because they too are subject to the neurological and biomechanical processes that influence written forms. However, as forms that instantiate quantity, they are also uniquely subject to the constraints of the perceptual system for quantity. As was mentioned earlier, signs instantiated by amounts that are subitizable tend to be highly conserved over time (Fig. 13.6). Amounts that are not subitizable are either replaced by conventions that mean bundled values or grouped as subitizable subgroups; over time, the latter tend to simplify to ciphered forms. Signs for numbers, particularly those that are not subitizable, are subject to the same neurological and biomechanical processes that influence written forms generally; however, signs for numbers remain identifiable as numbers because of the inherent numerical relations (e.g., unbundled maximums). This is why 35

Overmann, 2016a, 2021a.

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     numerical signs can be identified even in otherwise undeciphered scripts and unknown languages, like Linear A and Proto-Elamite.36 This visual change in form is analogous to the one that occurs in spoken numbers, where long descriptive phrases meaning as many as the fingers on one hand eventually become short unanalyzable words like five. The effects of numerosity on notations can be further mitigated through the use of additional bundles; an example is the V in Roman numerals, indicating the sub-base of five. The sign means that six is VI, rather than being parsed as subitizable subgroups (like it was in Mesopotamian cuneiform and Egyptian hieroglyphs). Numerosity also explains why IIII and VIIII, the original Roman forms, eventually became IV and IX, respectively, thereby reducing the four strokes that made them more difficult to appreciate visually. While the primary form used for written numbers remains the phonetically unspecified one (“7” and not “seven”), phonetically specified versions will eventually emerge. In Mesopotamia, phonetically specified small numbers did not emerge until several centuries after writing began, while phonetically specified large numbers did not develop for over a thousand years.37 The time lag between phonetically unspecified and specified numbers can be explained as the result of three things needing to be in place. Obviously, writing had to be invented first; remember that numbers had a much longer prehistory involving tokens, tallies, and fingers. Then techniques for phonetic specification had to be developed, as motivated by the need to reduce the ambiguity of the small pictures and conventions used for nonnumerical language. Critically, there also had to be a reason to apply these techniques to numbers, since numerical signs work better without the visual clutter that phonetic clues add. Eventually, reasons for phonetically specifying numbers did emerge. Speakers of Eblaite – which, like Akkadian, was an East Semitic language used in what is today Syria – recorded the sound values of Sumerian spoken numbers in the late-third millennium 36

Packard, 1974; Englund, 1998a, 2004; Corazza et al., 2021.

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37

Damerow, 1988.

  

. . Administrative tablet (TM.75.G.2198) from Ebla, modern Syria, circa 2350–2250 BCE. The obverse face contains the Sumerian words two through four and six through ten in phonetic form; one and five are not phonetically specified. While eight is said not to be analyzable as five [and] three (e.g., see Edzard, 1980, 2005), in this case, it clearly starts with the same signs as seven (which is analyzable as five [and] two) and is transliterated with the word for three. Photograph by M. Necci and drawing from Edzard (1980, Fig. 26a-b). The transliterations were taken from Pettinato (1981c, p. 212). Image © Missione Archeologica Italiana in Siria and used with permission.

BCE. An example is TM.75.G.2198 (Fig. 13.8), a third-millennium BCE tablet from Ebla. Its context suggests that Eblaite scribes wanted to learn the Sumerian spoken numbers in addition to the Sumerian numerical notations.38 In modern terms, it would be like our learning the Latin words for numbers in addition to Roman numerals, something that is not at all necessary for using the notations. Perhaps knowing the number-words of the classical language was considered prestigious, a hallmark of learning for a scribe. The artifact’s dating (2350–2250 BCE) is significant because it is also the time when the cuneiform system of writing first began to be adapted to the Akkadian language. It is possible that writing the sound values of the Sumerian number-words was an initial means and motivation for writing Akkadian words in cuneiform.39 There are a few exceptions to the rule that numerical notations begin as instantiated forms comprised of countable elements. One is found in Mesopotamia, where the Akkadians invented phonetically 38 39

Edzard, 1980; Pettinato, 1981a, 1981b, 1981c; Friberg, 1986. Overmann, 2021g, 2021a.

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     specified forms to represent decimal values like hundred and thousand that were productive in their indigenous decimal system but not concisely

represented

by

Sumerian

sexagesimal

notations

(Table 13.1): Another exception is the system of numerical symbols invented by the Amazonian Desana, probably during the nineteenth century, pursuant to their exposure to the idea of writing and Western decimal numbers.40 Both the Akkadian and Desana exceptions can be explained as motivated and potentialized differently than what is typical in numerical realization and elaboration. The Akkadians were highly numerate and literate, but their indigenous spoken numbers differed significantly from the foreign (Sumerian) notational system used in commerce and trade. Why the Akkadians did not invent nonphonetic conventions for these values is unknown; the use of borrowed notations highlights their lack of indigenous notations generally. The phonetic forms for hundred and thousand are no more succinct than their sexagesimal counterparts, and in the case of hundred, they are certainly more complex. The phoneticized forms were nonetheless preferred, despite their greater complexity. The compelling motivation was perhaps the desire to create something that was identifiably and uniquely Akkadian, a matter of cultural identity. Similarly, after the Desana were exposed to the idea of writing and decimal numbers in the nineteenth century, they adapted the foreign (Western) ideas in ways that reflected their indigenous culture.41 These symbols are not constructed with the accumulative, tokenized logic of the tally, in which each element is visually indistinguishable and represents another of whatever is being counted. Rather, some of the symbols represent cultural ideas and conventions. For the signs that do accumulate elements, each element is individuated, and its addition to a unique symbol creates another unique symbol that must be memorized. 40

Fernandes & Fernandes, 2006; Overmann et al., 2022.

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41

Overmann et al., 2022.

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Table 13.1 Sumerian nonphonetic signs for 100 and 1000 (top) and Akkadian phonetic notations for hundred and thousand (bottom). In the literature, many notations are transliterated phonetically (i.e., geš2 means sixty, not 60), showing that numerals are typically thought of as words. Nonphonetic transliterations are more accurate: The Sumerian sign for 100 is 1(60) 4(10), not 1(geš2) 4(u). Data from Grégoire (1996), Günbatti (1997), and Cavigneaux and Clevenstine (2018); also see Thureau-Dangin (1939). Akkadian cuneiform signs by Klaus Wagensonner. Numerical value Writing system

Characteristic

Sumerian

Written: Transliteration: Meaning: Artifact/line: Time period:

Akkadian

100 / hundred

1000 / thousand

1(geš2) 4(u) (1  60) + (4  10) Ashm 1924-1056, Obv. 1.7 2100–2000 BCE

1(geš-u) 6(geš2) 4(u) (1  600) + (6  60) + (4  10) MAH 15887, Obv. 6’ 1900–1600 BCE

1(diš) me-et 1 hundred Kt. j/k 97, Rev. 8 1950–1850 BCE

1(diš) li-im 1 thousand Kt. j/k 97, Obv. 25 1950–1850 BCE

Written: Transliteration: Meaning: Artifact/line: Time period:



     When it comes to numbers, these exceptions highlight that while it is possible to do things differently, doing so requires specific motivations, and outcomes reflect specific contexts, available resources, and cultural identity.

  –    –    Numeracy and literacy are severable phenomena, in more ways than the neurological dissociability noted in Chapter 3. Numeracy, after all, is reasoning with numbers, which can be written but do not have to be, so numbers have an extensive prehistory involving unwritten forms like fingers and tallies long before they become written. Literacy, on the other hand, is being able to produce and recognize written characters that express language, so it assumes a material form only with the first handwritten sign. As discussed, signs for numbers instantiate their meaning, a functionality that they share with their unwritten precursors; in comparison, signs for nonnumerical language signify, a mode whose ambiguity motivates the inclusion of clues that make the signs visually complex. Given this severability, it should not be surprising to learn that there are systems of numerical notations without nonnumerical writing, and a few systems of writing without numbers.42 Despite the severability of numeracy and literacy, when numbers and writing do co-occur, writing has several interesting effects on numbers, arguably yielding the qualities that make numbers the concepts that we know them as today. Certainly, the societies that had both numbers and writing – Mesopotamia, Egypt, China, and Mesoamerica43 – were also the ancient world’s great mathematical traditions. Here we want to examine the several ways in which writing – and not just writing for numbers, but writing for nonnumerical language as well – helps elaborate a system of numbers as a complex mathematical system. 42

Chrisomalis, 2010.

43

Senner, 1989; Powell, 2009.

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   First, when they are written, numbers are influenced toward becoming entities in their own right, rather than remaining as collections of discrete objects in the way that unwritten predecessors are. Numbers that are not written – seven Mesopotamian tokens, seven Yoruba cowries, seven Polynesian coconuts, seven Inka knots – are collections of discrete objects. In contrast, a handwritten number like the cuneiform seven (

) becomes recognized as an object in its own

right through the neurological reorganizations associated with handwriting and literacy. As written objects recognized through training effects in the fusiform gyrus, such signs were more likely to have been conceived as coherent entities than collections, even though they too consisted of multiple elements (e.g., like the cuneiform seven consists of seven wedges). This is not to suggest that numbers represented by unwritten material forms like fingers and tallies are always and only conceptualized as collections and never as entities. Certainly, language would help influence numbers toward entitivity, being conceptualized as an entity, since it is possible to say “there are seven” in a way that seven is not purely adjectival. However, we learn to recognize handwritten signs like we do physical objects. By training the fusiform gyrus to recognize written numbers as objects, written numbers would become objects, even though they were still composed of multiple elements in the same way their unwritten precursors were. Reconceptualization as coherent, semantically meaningful objects in their own right then supported the morphological change that eventually yielded ciphered forms. Second, written numbers are concise to a degree their precursors are not. A table of multiplication, for example, was not feasible with tokens or even proto-cuneiform notations in the way it was with cuneiform numerals; this is an area in which forms like Yoruba cowries, Polynesian coconuts, and Inka knots are particularly disadvantaged. Because they are so concise, notations significantly increase the amount of numerical information that can be brought together for simultaneous visual inspection. Conciseness afforded the collection

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    

. . Table of reciprocals of 60 (ERM 14645), provenience unknown, dated to the Old Babylonian period (1900–1600 BCE). (Right) Photograph of obverse face (cdli P211991); (left) translation based on Friberg (2009) and Friberg and Al-Rawi (2016). Tablets like these collected numerical relations at volumes that were not feasible with technologies like tokens or proto-cuneiform notations. Image courtesy of the State Hermitage Museum, St. Petersburg, Russia and used with permission.

of relations like multiplication into tables, representing relational information at an unprecedented volume (Fig. 13.9). Scribes created and learned tables of relations as part of their training, something we know through the schoolhouse texts and other records they left behind.44 The increased availability and accessibility of numerical relations made possible by writing and tables, along with the mandate to learn the relations, helped numbers to be reconceptualized, not just as objects, but as objects related numerically to one another. They became entities in a relational system.

44

Robson, 2003; Proust et al., 2007; Postgate, 2013.

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   Third, writing, tables, and relations gave scribes new options in calculating.

Certainly,

they

could

still

use

tokens,

whose

manipulability made them suitable for calculating in a way that notations were not because notations are fixed. We tend not to think of numerical notations in this way, as their fixedness is obscured by our use of interactive strategies that cross out and rewrite 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. Yet the notational calculation that we use required the development of the requisite relations, algorithms for manipulating the relations, and the preference for calculating by means of mental knowledge rather than physical manipulation. These would emerge gradually over the next several millennia, as evidenced by discussions in medieval Europe regarding the respective merits of the abacus and algorithms.45 But once tables of relations were available to them, scribes had the option to recall relations from memory or look them up in tables, and this intensified the transition to knowledge-based

calculation,

as

opposed

to

movement-based

calculation. Fourth, the ability to write nonnumerical language was crucial to mathematical elaboration. Just being able to write the results of calculations let scribes visualize and apprehend relations between wholes and parts in ways that had not been possible with earlier technologies, since rearranging things like tokens into parts destroys the whole, and tally notches cannot be rearranged. But as writing became more expressive, it was also used to record calculations, not just as results, as had been true with earlier technologies, and not in the form of equations, like we would do. Instead, calculations were recorded as narrative descriptions. 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.’).”46 While it would take several more millennia to develop concise, nonphonetic signs for even 45

Pullan, 1968; Stone, 1972; Evans, 1977; Reynolds, 1993.

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46

Devlin, 2003, p. 1.

     basic arithmetic (like our familiar plus and minus signs), these early narrative descriptions began the process by enabling calculations to be analyzed and to become more complex.

  Conciseness increases across the sequence of material forms, one of the several major trends that become apparent as new material forms are added to the cognitive system for numbers. For example, it takes the hands of eight people to represent the number 75, but only a single tally with 75 notches on it. Because they group, only seven Mesopotamian tokens would be required: a large cone, a small sphere, and five small cones. Continuing the reduction, only three cuneiform numerals would be required: the signs for sixty, ten, and five. In Western numerals, just two signs are needed: 75. Note that all these forms distribute the information across multiple material forms – 75 fingers on eight people, 75 marks on one tally, seven clay tokens, and two Western numerals – but the amount of physical material involved in each representation decreases, relative to what preceded it. The amount of physical material decreases because of another major trend, an increase in the knowledge the user must supply. Relatively little knowledge is needed to understand the hands of eight people or the requisite number of tally notches as representing 75. In comparison, forms that group require the user to know the conventions for any bundled values, as well as how the values are to be combined. For tokens, the user needed to know that a large cone represented sixty, a small sphere represented ten, and a small cone represented one, and that these values would be added together to achieve the total. Western numerals have the greatest implicit component. Not only must the user know the values represented by the ciphered signs 7 and 5, she must also know the meaning implied by the position of the numerals with respect to one another (as 7 is in the tens place and 5 is in the units place); their multiplication by the power of ten that position implies (so 7 is multiplied by 10 and 5 is multiplied by 1); and the addition that produces the final result (70 plus 5 yielding 75). The user must also have

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   acquired the neurological and behavioral reorganizations needed to engage the system of writing. Increasing the implicit component enables the explicit representation to become more concise. The implicit component also means that numerical concepts are not simply reducible to the material forms used to represent and manipulate them. Consider, for example, the four dots written or drawn in sand used by the Greeks to represent the number four and its internal relations;47 the four dots enabled four to be visualized as the literal square of two. The Mesopotamians using four small spheres, a highly similar form, would have understood forty, as each small sphere meant ten; the associated concept would have been a collection of discrete objects added together to achieve a total. The third major trend is distribution and independence: As new material forms are added to the cognitive system for numbers, numbers as concepts become distributed over multiple forms, in such a way that they become functionally independent of any one form. As we saw in Chapter 7, numbers do not have to be distributed or independent. Desana numbers, for example, are tied to the fingers in a way that Western numbers are not tied to numerals. Distribution and independence tend to make the material component invisible, with the result that if we think of the material forms used for numbers at all, we consider them as passive recipients for numbers that were mentally constructed. These trends make numbers in cultural systems differ in ways that we tend not to see. Here is an interesting historical example: When the Spanish first encountered Rapa Nui (dubbed Easter Island by early Western seafarers) in 1770, they compiled a list of words obtained from the islanders by touching body parts like arms and legs, pointing to objects like the large stone statues dotting the coastline, and pantomiming actions like standing up and eating.48 When it came to numbers, however, the Spaniards apparently displayed written 47

Klein, 1992.

48

González de Haedo, 1770.

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     signs for the numbers 1 through 10. Not unexpectedly, the words these elicited bore no resemblance whatsoever to the Austronesian lexical numbers used throughout Polynesia, whose use on Rapa Nui would be documented a mere four years later during the first expedition of James Cook.49 Today, there is consensus that “the Spaniards who compiled this list erroneously identified as numerals words which never were numerals.”50 In the historical documentation available today, there is little evidence of the Polynesian methods of tally-counting and the associated elaborate counting sequences on Rapa Nui, the settlement at the region’s easternmost extent.51 There are possible signs of forgetting during colonization, like the development of a unique (replacement) term for thousand, piere.52 However, it also seems highly unlikely that the counting sequences and methods themselves would have been completely lost – if for no other reason than the same vocabulary and decimal structure found throughout Polynesia is known for Rapa Nui, and the counting method would have had a role in preserving both. As a result, it is more parsimonious to assume that the islanders had the same highly elaborated system of numbers documented throughout Polynesia. So, when the Spanish and islanders met, two highly elaborated cultural systems of numbers were brought into contact: one that was mediated by written notations, a highly concise form with specific elaborational effects on the system of numbers, and one that was not. And the Spanish did something that is still very common today: assume that their cultural numbers represented all numbers. The Spaniards were not wrong in thinking their written notations meant numbers, only about what was involved in accessing their meaning as such. If they had perhaps displayed a tally instead, the islanders might have recognized its numerical function immediately. For one thing, the islanders used such devices themselves.53 But

49 51 53

50 Cook, 1777. Fedorova, 1993, p. 54. Overmann, 2020a, 2020b, 2020c, 2021c. Routledge, 1920.

52

Englert, 1977.

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   a tally also represents numbers with a significantly smaller implicit component. Notches lack, for example, the encipherment and positionality assumed in the notational forms that the Spaniards used. Using a tally also does not require the kind of neurological and behavioral reorganizations involved in learning to read. Similarly, if the Spanish had shown an abacus, the islanders might have recognized its function and organization as similar to their method of counting by sorting. However, in lacking both the implicit knowledge and neurological reorganizations needed to understand written notations as numbers, there was little possibility that the islanders would have recognized them as such. 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 or counting board. From the perspective of people enculturated into highly elaborated numbers like the Western cultural tradition, precursor technologies represent a subset of existing numerical knowledge, and are apprehended on that basis. The reverse, however, is untrue. From the perspective of people enculturated into number systems with relatively little elaboration, like Mundurukú, Desana, or Oksapmin numbers, 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. On the other hand, people with even restricted numbers are likely to understand the pragmatic realities of counting physical objects, something that has been lost in notationally mediated numbers. We will examine this in the final chapter. Also in the final chapter, we will discuss the so-called abstract– concrete distinction, the idea that some number concepts are attached to what they count in a way that others are not; here the concepts of distribution and independence will take a central role in how the distinction might be approached. We will also step back and look more broadly at what the role of material forms in numerical cognition might mean for both human cognition and numbers as concepts.

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 The Materiality of Numbers

In Book 7 of his famous Historíai, the Greek historian Herodotus wrote about Xerxes I, the king who in 480 BCE was mounting the second Persian invasion of Greece and would shortly fight the famous Battle of Thermopylae. But first, in an exceedingly odd footnote to history, Xerxes apparently needed to count his men, so when he came to a vast coastal plain in Thrace, a region that today overlaps the modern countries of Bulgaria, Turkey, and Greece, he halted his army:

[T]he number of troops in the whole land army added together was found to be 1,700,000. This is how they managed to count them. They gathered groups of 10,000 men together at one spot, packed them in as closely as they could, and then drew a circle around them from the outside. After delineating the circle and dismissing those 10,000, they erected a dry wall on the edge of the circle high enough to reach a man’s navel. When that was done, they had others go into the enclosure they had built, until they had counted them all in this manner. (Herodotus 7.60, translated in Strassler, 2007, p. 523)

Ever since Herodotus wrote this passage, scholars have debated the accuracy of his numbers, if not the situation itself. They have been skeptical of the purported size of the infantry, and while admitting possible deficiencies in the army’s ability to report its true strength,1 they have scoffed at the idea that the king and his commanders would not know the number of their troops with reasonable accuracy unless they first gathered them into a single place and counted them.2 Some scholars have highlighted the situation’s infeasibility, either on

1

Maurice, 1930, pp. 233–234, note 54.

2

Macan, 1908b, p. 155.



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     grounds of the amount of time required3 or the negative impacts such an endeavor would have on the military enterprise,4 as an army idled for any purpose still needs to eat, and in the meantime, it is not fighting. A few have focused on reasons why the sums were so clearly exaggerated. Perhaps they were meant to please the king,5 flattering him as so very mighty indeed as to command such a powerful force, or intending to frighten their enemies by reporting an army so overwhelmingly massive. One, observing that the enumeration process penned up men like so many sheep for slaughter, read the passage as a sad commentary on the pathos of war.6 Another suggested that Herodotus had made an unfortunate mistake in numerical translation, confusing chiliad, the Persian unit for a thousand men, with myriad, the Greek unit for ten times that number.7 The idea that Herodotus simply mistranslated the numbers is almost certainly wrong, since more than a century later, another Persian king, Darius III, would count his men in exactly the same way, as reported by the Roman historian Quintus Curtius Rufus:

Accordingly, having encamped before Babylon, [Darius] made a display of all his forces, in order that they might enter upon the war with the greater confidence, and having built a circular enclosure, capable of containing a throng of 10,000 armed men, he began to number them as Xerxes had done. From sunrise to nightfall the troops entered the enclosure, as they had been told off. (Quintus Curtius Rufus 3.2–3, translated in Rolfe, 1946, p. 73)

Presumably, the enumeration task was somewhat easier this second time around, since Darius’ total force was only about a fifth of the 1.7 million men that Xerxes had supposedly mustered in just his infantry alone. Undoubtedly, collecting men into an enclosure big enough to contain 10,000 of them at a time – and repeating this

3 6

4 5 Macan, 1908a, p. 82. Flower, 2007, p. 819. Grote, 1851, p. 36. 7 Crosby, 1860, p. 271, note 2. Flower, 2007, p. 821.

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     procedure more than 30 times to count the army of Darius, 170 times for the infantry of Xerxes – remains a particularly awkward and unusual way to count soldiers. This is true even when the quantities in question are smaller, since all the aforementioned constraints and caveats discussed for Xerxes’ army would still apply, if to lesser extents because of the smaller number of Darius’ men. So why the “rude” counting method?8 Scholars have long suspected an inability to comprehend very high numbers.9 Yet the number systems these kings would have known could count into the millions, and they had achieved the status of a complex mathematical tradition more than a thousand years earlier. Thus, numerical incompetence is hardly likely to have been the case. This would be especially true for men like kings who were highly educated and routinely dealt with matters of trade and military logistics, endeavors for which the ability to comprehend numbers had been crucial for even longer periods of time. Let us grant that the armies may have lacked the ability to report their strength with the desirable accuracy and alacrity;10 it still does not follow that they would resort to such crude measures in order to count their men. Let us grant instead that ancient peoples were skilled and efficient at counting physical items of any sort, including soldiers. On this supposition, a new explanation appears: Xerxes and Darius had so many men, they could not count them in the usual way, but instead had to measure them as if they were grain – by the number of times the men could fill a container holding 10,000 at a time.11 This would be consistent with the other boasts Xerxes is reported to have made: His men were so numerous, they could drink a river dry; his men were so numerous, they would bankrupt in a day any city that tried to feed them.12 Our failure to see this footnote to history as more of the same royal puffery shows just how far divorced our Western numbers have

8 10 12

9 Grote, 1851, p. 34, note 2. Flower, 2007, p. 823. 11 Maurice, 1930, pp. 233–234, note 54. Macan, 1908a p. 82, note 4(i). Flower, 2007, p. 819.

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     become from the practical realities of counting physical objects. This is an area where the role of materiality is both particularly pronounced and almost totally invisible and misunderstood. It is pronounced in that material forms – not just Mesopotamian tokens and notations, but Inka knots and Yoruba cowries – represent things that are being counted, but also remove them from the counting process. Essentially, counting tokens or cowries that represent things like sacks of grain or skeins of wool is not the same as counting the sacks of grain or skeins of wool themselves. This is the kind of abstraction at issue in the so-called distinction between numbers that are abstract and numbers that are concrete:

Numbers are divided into two classes, Abstract and Concrete. When they are applied to particular objects, as peaches, pounds, yards, &c., they are called concrete. When they are not applied to any particular object, they are called abstract. . . . Thus, when it is said that two and three are five, the two, three, and five denote abstract numbers. (Thomson, 1846, p. 261)

Historically, the abstract–concrete distinction has tried to differentiate Western numbers, which are presumed to be abstract, from non-Western ones, which are presumed concrete. This characterization emerged from nineteenth-century ideas about the way societal thinking develops and progresses, as we saw was the case with applications of Piaget’s developmental theory to entire societies. Certainly, beyond differences in structure and organization are often differences in elaboration between Western and non-Western numbers. However, the abstract–concrete distinction is a poor and misleading way to discuss them: It characterizes any differences pejoratively, incorrectly simplifies difference to a binary state, and fails to distinguish how and why numerical concepts might change in terms of their content, structure, and organization. It also ignores the fact that numbers are abstract from their inception: Using the twoness of arms to characterize twoness in other

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     pairs, as the Mundurukú are reported to do,13 not only demonstrates connections between sets of objects, Russell’s definition of a number,14 it also exemplifies the psychological construct of abstraction. Thus, numbers do not become abstract; rather, they are always abstract. Thinking of numbers in terms of abstractness and concreteness misunderstands their nature as concepts, which is always a mix of neural activity, material forms instantiating numerical information, and behaviors interfacing the two, independent of the degree of numerical elaboration that might characterize a particular cultural system. The role of materiality in numbers is invisible and misunderstood because, most curiously, the abstract–concrete distinction always focuses on the material forms that numbers are used to count. It never considers the material forms that represent or manipulate numbers, unless these have an unusual form, like the 10,000-men container said to have been employed by the ancient Persian kings. This is because our Western numbers have become distributed over so many different material forms – written numerals and the fingers; tally marks and manipulable forms like coins or an abacus; sometimes Roman numerals or binary and hexadecimal numbers – that they have become independent of any particular form, to such an extent that we no longer think of them as materially realized or materially bound. The formal devices and ad hoc forms that we use for representation and manipulation have disappeared entirely from the way we think about numbers. This is a version of Wittgenstein’s ladder, a conceit of the Austrian philosopher Ludwig Wittgenstein: Once we master a conceptual system, our understanding of it is such that we have, so to speak, climbed to its top and pulled up after us the ladder we have just used to ascend.15 In the case of our Western numbers, now that they have become so highly elaborated, the various devices that once gave them their properties have become not just invisible to us, but are in fact misunderstood once they are pointed out.

13 15

Pica & Lecomte, 2008; Rooryck et al., 2017. Wittgenstein, 1933.

14

Russell, 1920.

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     The idea that concrete numbers are conceptually connected to the objects they count is difficult to reconcile with Mundurukú numbers, for if they were so connected, the only thing countable with a concept of twoness based on the arms might be another pair of arms. The idea of concreteness as defined by what numbers are used to count is also difficult to reconcile with Desana numbers, which are overtly tied to the fingers that materially realize them, at least as closely as they might be to any of the things that they might be used to count. Both number systems involve material forms, though the fingers used by the Desana have an advantage, relative to distributed exemplars like the arms used by the Mundurukú for two, in being a contiguous material form that adds structure and organization to the conceptual system. Interestingly, the Polynesian method of counting by sorting collapses any difference between the object being counted and the material form being used to keep track of the count. This suggests an increased technological distance to a material form that would help the conceptual system to elaborate further; perhaps it might effectively preclude the material and behavioral abstraction that would otherwise occur when counting is performed with proxies like cowries, tokens, pebbles, beads, sticks, or kernels of corn, and in places away from the enumerated goods themselves. Unfortunately, there has been almost no cross-cultural research into this area, beyond noting overt differences of structure and organization (e.g., decimal vs. vigesimal) between number systems and correlating their extent (Greenberg’s limiting number16) with the potential for relations between numbers.17 But then, there has been almost no investigation of the role of materiality in conceptualizing numbers more broadly speaking. This is most likely explainable as the result of assuming numbers to be wholly mental creations whose monolithic ideal resembles their Western incarnation.

16

Greenberg, 1978.

17

Bender & Beller, 2006.

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     Another, more recent example in which an unfamiliar and arguably material counting practice confronts our blindness to the materiality of numbers comes from Africa. In 1862, the Paris-based missionary Christian Schrumpf observed that in Sessuto (or Sesotho), a language spoken across much of the southernmost part of Africa, counting over one hundred required the combined efforts of three men:

The adjectival numerals in the Sessuto language are extensive and somewhat awkward. That is why counting, when the number of objects to be counted is considerable, is almost a huge thing for the natives. When counting beyond a hundred, three men usually have to do the difficult work together. One man counts the units on his fingers, which he raises one after the other, thereby indicating the object counted or, if possible, touching it. The second man raises his fingers for the tens (always starting with the little finger of the left hand and continuing to the little finger of the right hand) as they are completed. The third man counts the hundreds. (Schrumpf, 1862, pp. 463–464, as translated)

Schrumpf and other authors interpreted this behavior as indicating a particular awkwardness with numbers, perhaps the “difficulty of visualizing larger numbers by using the fingers only.”18 It is true that Sessuto names for higher numbers are lengthy, and that the longer the names in any language, the less well suited they are for being recited in sequence to count things. Just counting from fifteen to twenty might seem a chore, especially for those who, like Schrumpf, were accustomed to names of fewer syllables: leshume le metso e metlanu (“a ten which has five roots”), leshume le metso e tseletseng, leshume le metso e shupileng, leshume le metso e robileng menomeli, leshume le metso e robileng mono o le mong, mashume a le mubeli (“two tens”).19 The same numbers in the

18

Cajori, 1899, p. 35.

19

Schrumpf, 1862, pp. 464–465; as glossed in Matsau, 2009.

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     German Schrumpf was writing in are fünfzehn, sechzehn, siebzehn, achtzehn, neunzehn, and zwanzig. Why an awkwardness with numbers? Counting to quantities higher than ten with the fingers means keeping track of the tens and their multiples in some fashion, in addition to keeping track of the units. Keeping track would presumably involve the names for numbers, and hence, be a matter in which particularly lengthy expressions would prove awkward. And if collaborative rather than individual counting was the norm, any particular individual enculturated into such a system might find it difficult to count sequentially from one to one hundred by himself – assuming it would ever occur to him to deviate from the customary counting method. Yet thinking that the counting method simply represents awkwardness entirely misses the larger point: By separately keeping track of the units, tens, and hundreds, the men had created an abacus. If the device was more suited for accumulation than calculation, the result was nonetheless a positional system, one in which each man’s position literally determined the value of the numbers he was tracking.20 Moreover, the system’s exponential structure was theoretically extensible to very high numbers, since a fourth man could keep track of the thousands, a fifth the ten thousands, and so on. But regardless of how many exponents were involved, each man’s task was simple and straightforward: He only had to count from one to ten and keep track of this count with his fingers. Moreover, most of the Sessuto words in this range are quite short, and using the fingers would increase the accuracy and reliability of counting by decreasing the demands on human memory. And the exponential structure of numbers would be much clearer in this collaborative abacus-like counting, where a number like 888 would be expressed as eight understood in the hundreds tracked by the third man, eight understood in the tens tracked by the second man, and eight

20

Gerdes, 2008, p. 1762.

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     understood in the units tracked by the first man. In contrast, in individual sequential counting:

[I]f you want to express 888 (which already has a good length in German) in Sessuto, you have to work through the following sentences: makholu a roblieng meno o le meli a nang le mashume a robileng meno meli, le metso e robileng meno e le meli (hundreds they are broken, which fingers, they are two, which are with tens they are broken, which fingers are two, and roots (units) they are broken, which they [i.e., fingers] are two). (Schrumpf, 1862, p. 464, as translated)21

Other benefits might accrue from collaboratively counting in the Sessuto fashion. Sequential recitation, even just the words from one to ten, would put any longer phrases under pressure to shorten. Eventually, this would truncate longer phrases as short, unanalyzable number-words.22 Such collaboration would also expose participants and witnesses alike to the naming of very high numbers by means of just the words for one through ten and any exponential multipliers like tens and hundreds. Such exposure might do more than shorten the names: It might cause them to become more regular, and the process of naming to become lexicalized as well, in the same way discussed for Polynesian counting in Chapter 12. In terms of future directions for cross-cultural studies of number systems, several things emerge from this discussion: First, we should expose and then discard the assumption that numbers are a mentally created monolithic construct whose result is isomorphic, beyond a few inconsequential surface details, with Western numbers. Second, we should document cultural number systems, not just in terms of 21

22

I am indebted to my friend and colleague Annick Payne for her help in translating this passage; the result faithfully represents Schrumpf’s (1862, p. 464) awkward translation of the original Sessuto phrase into antiquated German: “(Hunderte sie sind gebrochen, welche Finger, sie sind zwei, die sind mit Zehner sie sind gebrochen, welche Finger zwei, und Wurzeln (Einlieiten) sie sind gebrochen welche sie sind zwei).” Schrumpf, 1862, p. 464.

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     things like structure, organization, extent, and verbal forms, but also whether material devices are used, which ones are used, exactly how they are used, and how these matters correlate with numerical properties. Third, we should see counting as a pragmatic process that involves physical objects, and which is mediated by both the things counted and the things used to count, and then investigate how these matters affect how numbers are conceptualized. Fourth, we need to create ways for expressing elaborational differences that are more nuanced than the abstract–concrete distinction; we need ways that recognize numbers as abstract from their inception and materially bound no matter how elaborated they become.

    If we are to gain new insights into numeracy, both prehistoric and contemporary, we need to understand much more about how material forms contribute to it, and this is a domain that archaeology, as the science of material culture, is uniquely equipped to answer. We need to question the rationale for including only some of the material forms used for numbers – the ones that happen to be convenient in terms of their identification and analysis as possible devices for counting – while concomitantly excluding other forms that are manuovisually engaged but not traditionally material. For archaeology, this means all the things attested ethnographically that would not preserve; for the cognitive sciences, it means including all the things engaged manuovisually, not just written notations and subitizable nonsymbolic quantities of objects. Broadening our scope means using ethnographic data for insight into which material forms are used in numbers and how they are used. The insights gained from this should then guide what we look for in the prehistoric material record and how we interpret what we find there.23 Assuming that prehistoric and contemporary numbers represent a single realization process and not several, the contemporary 23

Overmann, 2023b.

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     tendency to use ephemeral and perishable forms implies that the prehistoric record is unlikely to contain evidence of the first steps into numbers. Relatedly, prehistoric numeracy is likely to be misinterpreted if assessments are based solely on the material record, since what it does contain either represents a subsequent, more elaborated state, or things that are not numbers. This means that new techniques for estimating societal numeracy are needed. One such might try to infer prehistoric societal numeracy from socio-material conditions; this could be anchored in detailed analyses of how these variables correlate in contemporary societies. Analytical techniques will also need to be developed for numerical objects that are not notched bones and stones, something that currently available methods do not and currently cannot do; for differentiating notations from nonnumerical marks, in the ways that Hayden’s recent criteria have advanced;24 and for differentiating numbers from other notations. Underlying artifactual interpretations is the assumption that we know what we are looking for when we look for numbers in the material record. Numerical traditions often become more elaborated over time, but they can also persist unchanged, decline, and go extinct. Numerical traditions also differ cross-culturally in attributes like extent, organization, relations between numbers, and social purposes. Today, the greatest variability between cultural number systems is found in places like the Amazon and Papua New Guinea, where number systems tend to be associated with small group sizes and conditions of relative isolation, and some are also thought to be either in the process of emerging or relatively recently emerged. Prehistoric number systems, in being associated with similar sociomaterial conditions and in being in the process of emerging or having recently emerged, are likely to be both less elaborated and significantly varied. And this means, assuming that we can develop reliable methods and criteria for discerning prehistoric number systems from

24

Hayden, 2021.

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     artifactual marks, that the very last numbers we should expect to find in the prehistoric record are Western ones.

   Scholars have been asking what numbers are and where they come from at least since the time of Plato, an impressive span of about 2500 years. To answer these questions, numbers have been investigated from a variety of perspectives: philosophical, historical, mathematical, logical, linguistical, psychological, neurological, comparative (animal studies), ethnographic, semiotic, notational, and archaeological, to name just the main ones. Arguably, satisfactory answers have yet to be found, judging by the fact that we are still looking for them. This may be because numbers have always been approached, in one form or another, as in-the-head phenomena. We have not previously recognized, let alone dealt with in a meaningful way, the material component in numerical cognition, the devices used for representing and manipulating numbers. After two-and-a-half millennia, it is more than time to address this lacuna – to look at the materiality of numbers. Where are numbers left – what position might they hold – when we consider them to be materially bound no matter how elaborated they become? When we consider them to elaborate by recruiting new material forms and taking on their properties? When we consider their conceptualization and thinking to involve material forms as an integral component of a cognition that is extended and enactive? On the one hand, the material view seems to make numbers something other than what we have always assumed them to be, ideas created by the human brain. On the other, seeing their unfamiliar material component illuminates something new, not just about number systems and their cross-cultural variability, but also about the nature of our cognition generally and the process whereby our conceptual systems change. These insights are useful, especially if we want to understand how we think and how our ability to think developed evolutionarily. This ability to leverage material forms for cognitive purposes is

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     perhaps the fundamental difference between ourselves and all other species. This material difference may include even our cousin species, the Neandertals, whose larger but differently shaped brains imply cognitive differences in domains like creativity, and whose material record suggests a slightly different and perhaps less creative relationship with material forms.25 Some interesting and difficult questions emerge from this: Are numbers different or special in the ways they involve materiality, or do other domains also elaborate by leveraging material forms for cognitive purposes? If material objects are a constitutive element of our individual cognitive systems, and we all have the same material form as an element of our individual minds, does it follow that our cognition is, to some extent, shared and collective? After all, we share similarities in our behavioral, psychological, physiological, morphological, and linguistic characteristics too, without our drawing any conclusion other than it makes sense that we should resemble one another in these ways because we are, after all, members of the same species and societies. Or is this just another way in which numbers are unusual or special, a reason for their strong universality? Numbers have properties that are intriguing from a cognitive perspective, like the unusual conviction with which we believe that 2 + 2 = 4 and that a mathematical proof is correct. As noted in earlier chapters, these things seem to exist in some way that is external to the individual mind. Possibly these phenomena are a result of numbers being born of shared material structures; possibly this quality of numbers also draws upon shared cognitive capacities innate to the human brain. In the latter case, there are currently a lot of unknowns: what those innate capacities might actually consist of, beyond numerosity; how they might influence material form and function in the historical development of numerical concepts; and whether they can be meaningfully separated from the material forms we use to represent and manipulate numbers. 25

Wynn et al., 2016.

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     The mathematician LEJ Brouwer seems to have envisioned numbers and our capacity for understanding them as reflections and products, not just of the human mind but also of the universe it inhabits. If we reimagine the human mind as something larger than the brain – something that directly and constitutively includes material aspects of the universe of which it is also a part – perhaps then we can begin to understand numbers, even beyond non-neurocentric and nonrepresentationalist models, as in some way directly comprehending the universe of which the mind is a part.

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Index

abacus, 335, 339 cowrie grid, 298, 304, 306–307 design, 284 ephemeral, 288, 292, 296–297, 319, 347 Roman, 14, 279 transition to algorithms, 19 use in display, 163 use in manipulating, 163 Abipónes, 13, 73, 77, 215 absolute dating, 102 abstract numbers. See abstract–concrete distinction abstract thinking, 30 abstract–concrete distinction, 37, 41, 297, 339, 343–344, 349 abstraction, 43, 55, 72, 343–345 behavioral and material forms of, 56 lack of focus on in psychology, 57 as purely mental activity, 56 acalculia, 62 accessibility, 273, 276 accumulation, 277 accumulation as central organizing principle, 222 accumulation and distribution of cognitive effort, 173, 176, 185, 192, 221 additive number system, 286 affordances, 150, 156, 186 of calculators, 187 encoded in material artifacts, 187 agency, 173, 176, 182, 185–187, 189, 220 agent, 156, 185–186 Aguarunas, 74 Aimoré, 77 Ainu, 14 Akkadian, 101–103, 109, 328–329 algorithms, 7, 19, 163, 187, 189, 282, 286, 307, 309, 335 alphabetic numbers, 321 Alyawarr, 96 amphibians, 50 analyzable forms, 84, 96, 101–102, 127–128, 140, 216

ancient migration arcs, 129 Andamanese, 71 angular gyrus, 61 animacy, 109 Arabic, 104, 108 archaeological evidence, time depth of, 41 archaeological survivability, 136 Aristotle, 3, 177 artifacts analyzed for possible use in numbers, 246 atoms, 85, 96–97, 99, 112, 114, 116, 120, 216 attention, 44–45, 326 Australian message sticks, 260–261, 263 functions and purposes, 260 resemblance to Upper Palaeolithic artifacts, 261 australopithecines, 247 Aymara, 79 back-counting, 319 Bakaïrí, 72, 91, 201, 214 Baniwa, 230, 233 Barasana, 272 base fourteen, 21 base number, anatomically derived, 21 Bashila, 75 beads from Bizmoune Cave, 245 from Blombos Cave, 8 characteristics of, 245 more likely ornaments than counting, 245 of the Pomo, 138, 140 from Skhul Cave, 245 from Sunghir, 138 techniques, shortalls in, 243, 245, 267 behavioral automaticity, 205 being and nonbeing, 19 Bergdama, 213 birds, 50 Bizmoune Cave. See beads from Bizmoune Cave blind man’s stick, 183–184



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  Blombos Cave, 237 body, instrumental use, 202 body-counting, 81, 98, 115, 207, 211, 222–223, 225, 228, 241–242, 244 bone from Border Cave, 8, 269 bone from Ishango, 253, 274, 304 bone from Les Pradelles, 253, 270 bones from the Levant and Zagros Mountains, 141 bone from Nesher Ramla, 267 bone from Taï, 272–273 Border Cave, 8 brain morphology, differences between human species, 64 brains, what they are very good at, 56, 169 Brothers Grimm, 212 Brouwer, LEJ, 27–31, 72, 353 building blocks, 76, 200–201, 216 bullae, 313 bundling/debundling, 283 calculation European Middle Ages, 164 knowledge-based, 5, 19, 163, 309, 335 movement-based, 5, 19, 335 calculators, 164, 187 capacity, 230 cardinality, 3, 92, 233 Cartesian dualism, 166, 178–179 categorizing, 43, 51, 54, 72 cross-cultural view of, 54 lack of focus on in psychology, 57 and stone tools, 53 cerebellum, 64, 66 new understanding of roles, 64 and the origin of mathematics, 64 role in creativity, 64–65 role in movement and functions important in numbers, 90 cerebrum, 64 China, 332 Chinese, 104, 106, 108, 112, 116, 215, 321 Chinese suanpan, 279 Chukchi, 77 clay impressions, 8, 140–141, 252, 266, 312, 314 linearity in, 160 coconuts, 288, 296, 308, 333 cognition, unconventional view of, xxi cognitive approach to material objects, 1

color, compared to quantity, 47, 73, 120 common use, 174–175 averaging effect of, 175 common change in behaviors and brains, 174 emphasis of material features, 174 synchronization with user abilities, 175 compounds, 83, 96–98, 101–102, 112, 114, 214 computers, 164 conceptual acquisition, xx, 36 conceptual blending, 149, 164, 167 materially anchored, 150, 167, 169 conciseness, 158, 276, 309, 311, 333, 336 concrete numbers. See abstract–concrete distinction concrete thinking, 30 congenitally blind people, lack of fingercounting in, 71, 86, 114, 202 congruent material and linguistic forms, 115 conservation rate in small number-words, 41, 99, 263 conventions, 311 Cosquer Cave. See hand stencils at Cosquer Cave parietal art, 219 counting boards, 283–284, 308 properties of, 169 counting by eights, 293 counting by elevens, 297 counting by fours, 292 counting by pairs, 292, 297 counting physical objects, pragmatic realities of, 289, 295–296, 298, 300, 339, 342, 349 counting sequence, 3 counting by sorting. See tally-counting counting by visual inspection alone, 290 counting yams in Papua New Guinea, 295 cowrie counting system, 298 cowrie shells, 298, 308, 333 creativity, 65, 150, 171, 272 creating in common, 192–193 cross-cultural variability, 16, 30, 87, 118–119, 151, 166, 351 cultural analogy, 143–145, 147, 239, 254, 257, 259, 270, 272, 274, 276, 349 and empirical verifiability, 239 cultural identity, 134, 330 cultural names, 35

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  cultural notational systems, gaps in the literature, 147, 246, 257, 259, 269–270, 276 cumulative-additive system, 319 cumulative-positional number system, 318 Cuneiform Digital Library, 107 cuneiform numbers, 266, 314 Darius III, 341, 344 decimal, 5, 14, 21, 209, 301, 330, 338 decontextualization, 55–56 demands on cognitive resources, 85, 88, 90, 98, 187, 190, 205, 216, 227, 289–290, 292, 347 Desana, 13, 78, 84, 135, 206, 222–223, 228–229, 231, 233, 235, 238, 240–241, 244, 254, 330, 337, 339, 345 Descartes, René, 29, 178 design space, 150, 159 determinatives, 326 developmental acquisition of numbers, 23, 39–40, 172 device capabilities and limitations, 149, 156, 158–160, 170, 188, 221–222, 309 diffusion, 125 digit-tally, 222, 228–229, 234, 241–242, 244, 254, 277 disconnected, mixed-base systems, 297 discoverability, 11, 27 discrete infinity, 119, 122 discreteness, 100, 187, 204 realized through language, 95 distinction, objects being counted and counting device, 296 distributed, as conceptual property, xix, 158, 172, 336–337, 344 distributed exemplars, 1, 134, 151, 156–157, 162, 204, 216, 345 double-enumeration strategy, 226 doubly dissociable, 44 duck or rabbit visual illusion, 200 duodecimal, 21 the dyad, 18 early writing, 174 for nonnumerical language, 174 for numbers, 174 Easter Island. See Rapa Nui Eblaite, 328 Egypt, 311, 332

El Castillo, 217 Elamite, 109–110, 142 embedded cognition, 179 embodied cognition, 179 embodied model, 32–33 enacted cognition, 176, 179, 182 enactive signification, 177, 190 enactive space, 149, 169, 183 enactivity, 169, 183–184 enculturation, 39, 57 English, 77, 96, 104, 106, 108–109, 116, 216, 301, 321 entitivity, 333 ephemerality, 203 ethnographic data, 70, 137, 143, 156, 158, 255, 257–259 historical, 143 reliability of the historical data, 35, 144 uniformist comparisons in, 144 etymological roots, 96 etymological transparent forms, 84, 127 etymologically opaque forms, 84 evolutionary change in behavior at 1.8 million years ago, 122–123 evolutionary history of tool use, 54 experimental marks, 191, 255 explicit component, 158, 337 extended cognition, 176, 179, 182–183 extended model, 32, 34 external memory storage, 42 external representations, constitutive role, 17 familiarity, 171 Fiji, 211 finger agnosia, 62 finger gnosia, 60, 62 finger-counting, 24, 87, 159, 199, 204, 217 age, 217 Bede’s system, 209 biomechanical constraints, 200 collaborative forms, 209–210, 224, 279 cross-cultural variability in, 71, 78 cultural beliefs about certain finger patterns, 201 fingers individually enumerated, 206 habit, 205 idea of handful, 206 impaired, 62 without individuating the fingers, 206 infeasible patterns, 80

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  keeping track of the number of cycles, 208 linearity, 87 neurological interconnections, 24, 86, 150, 202 neurologically predisposed, 66 not demonstrated by chimpanzees, 53, 202, 204 oldest linguistic traces of, 82 place in material chronology, 157 prevalence in numerical emergence, 235 properties, 170 role in discreteness, 87 self-reference, 208 as social behavior, 220 span of numerical elaboration, 88, 159 use in developmental acquisition, 172 visual choice, 200 visual experience of the hand, 86 what all cultural variants have in common, 88, 204 finger-montring, 80, 203, 206 fingers differentiated as material form, 71 the hand as distributed exemplar, 151, 215 the hand as instrument and actor, 67 hands as features of socio-cultural environments, 86 individuated, 215, 232, 235, 238 limited morphological plasticity of, 162, 220 material properties of, 202 names for number-words, 215 properties of, 169 typical basis of a number system, 76 use with spoken numbers, 75 used without language to express numbers, 75 first nonsubitizable numbers, 76 first numbers to emerge, 46, 70, 100 first unambiguous numbers, 8, 124, 129, 140, 263 fish, 49–50 fixed and suitable for recording, 149, 156 fixedness, 309 foundationalism, 28 free will, 185 Frege, Gottlob, 26, 145 frequency of use, 47, 84–85, 97–99, 102, 104, 106, 108, 114, 127, 129, 213, 216, 296

Fuegian, 71 functional fixedness, 171 fundamental assumptions about numbers, xix fusiform gyrus, 174, 324, 333 Gall, Franz Joseph, 178 gap numbers, 76 Gargas Cave. See hand stencils at Gargas Cave generalization, 55–56 geometry incorporating material forms, 196 as the material control of spatial perception, 195, 197–198 Gerstmann syndrome, 62 gesture, 70, 75, 85, 89, 129 evolutionary history, 123 as a form of language, 70 and memory, 90 precedes the verbal expression of numbers, 73 role in bringing forth meaning, 95 use in communicating between languages, 73 global pattern, 126, 129 global distribution of material technology, 125 Gooniyandi, 187 grammatical number, 23, 47, 108–109 development in conjunction with very few lexical numbers, 111 dual form in Semitic languages, 108 modulation by animacy, 109 trial form in Austronesian languages, 108 Greek philosophical ideas about numbers, 19 Greek use of four dots, 337 Grimm’s dictum, 212 grouping, 158, 266, 277 natural grouping of hands and feet, 205 Guamán Poma, 281–282, 284, 286 hand stencils interpreting patterns, 219 nonrandom distribution of finger patterns, 218 shortened digits, 217 techniques, shortalls in, 267 hand stencils at Cosquer Cave, 217–219, 246 hand stencils at Gargas Cave, 218

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  handwriting, neurological and behavioral reorganizations, 312 handwritten notations, 156, 309, 312, 320 as material form, 320 Herodotus, 340–341 highest number counted, 259 Hindu–Arabic notations, 5, 18 Hippasus, 19 Hixcaranya, 215 hominin, 54, 247 Homo erectus, 60, 125 Homo habilis, 247 Homo sapiens, 61, 125–126, 272 differences in material culture from Neandertals, 65 distinctive parietal encephalization, 61, 123 dramatic change in material culture, 124 evolutionary history of tool use, 90 globular brain shape, 61 larger cerebellar volume, 64 honeybees, 50, 123 Hup, 215 hylozoism, 179, 185 Igbo (or Ibo), 59, 298, 301, 308 imitation, role in finger-counting, 80 implicit component, 158, 160, 169–170, 312, 336–337, 339 implicit knowledge, 339, also see implicit component incongruent material and linguistic forms, 115 independence, xix, 158, 169, 172, 337, 344 individual variability, 193 Inka, xvii, 151, 164, 279, 282–283, 315, 333, 343 Inka Empire, 135, 137, 281, 316 Inka language, 137 Inka number system, 135, 281, 283–284 inner speech, 62 innovation, 132, 161, 171, 173, 193 cumulative, 162 insects, 50 instantiation, 182, 190, 309, 312, 314–315, 321–322 integers, 3 intelligibility, 276 intentionality, 186 intersubjective verifiability, 26, 352

intraparietal sulcus, 43, 59, 63 linkage with other regions, 62 regions specialized in, 61 role in learning ordinal sequences, 63 introspection, 29 intuitionism, 27, 33 Inuit, 209 Greenland, 77 Hudson Bay, 77 Iqwaye, 208 irrational numbers, 19 irregular compounds, 216 Asian language exceptions, 112 Ishango. See bone from Ishango iwakalua, meaning of explained, 293 Japanese soroban, 279 jetton, 19 Jibaros, 74 judgments of relations, 53 judgments of cross-dimensional relations, 53, 55 judgments of identity, 53–54 Kakoli, 14 Kapauku, 125 Karam, 208 khipu, xvii, 135, 164, 280–284, 315–316, 318–319 in the Amazon, 135, 230, 233, 317 not a device for computation, xvii numerical component, xvii use of perishable materials, 136 Kingdom of Dahomey, 303 Koryak, 77 La Trahison des Images, 322 language, 43 defined, 94 distinct neuroanatomy and neural activation patterns, 44 evolutionary emergence, 50, 122 lack of descriptiveness for numbers, 15 origin hypotheses, 122 well-formed ways of expressing numbers, 59 Lanì, 75 large, interconnected societies, 99, 130, 132, 146 Latin, 12, 14, 327, 329

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  Lebombo bone. See bone from Border Cave Les Pradelles. See bone from Les Pradelles Levallois technique, 65 Lévy-Bruhl, Lucien, 30, 37, 40, 56, 111 lexical numbers, 47, 96, 105, 112 lexical rules, 96–97, 216, 292 lexicalization, 216, 296 Linear A, 11, 328 linear continuum, 15 linearity, 188, 233 linguistic evidence, time depth of, 41 linguistic model, xxiii, 32–33, 95, 118 linguistic sign, 177, 190, 216 literacy, 174–175, 312, 332 Luiseño, 206, 208 magnitude appreciation, 45–47, 198 effects on material devices and language, 46 maize kernels, 282, 296, 308 mammals, 50 Mangareva, 293 manipulability, 195, 197, 220, 222, 227, 335 manipulation as central organizing principle, 223, 276 marks on the ground, 70 Marshall islanders, 279 material anchoring and stabilizing, 190 material chronology, 156–157, 162, 188, 309, 312 Material Engagement Theory, xxi, 176–177, 179, 191 material forms as active participants rather than passive repositories, xix, 17, 42, 114, 172, 197, 308, 337 anchoring and stabilizing concepts, 53 mobile and suitable for calculating, 149, 156 material heaviness and conceptual units of weight, 167 material sign, 177, 190, 216, 320 materiality, xviii, 1, 53, 169, 171, 173, 186, 190, 193, 343–345, 352 cumulative effects of using, 193 stretching the definition, xix transfer of conceptual structure between material forms, 171 mathematical proof, 11, 352 memory, 45 memory effects, 98

mental abacus, 89 mental computation, 89 importance of finger movements in, 89 mental lexicon, 85, 97–98, 120 mental number line, 15, 43, 58–59, 68 importance of visual exposure, 58 logarithmic or linear disposition, 58 Mesoamerica, 279, 332 Mesopotamia, 8, 21, 101, 104, 106, 109, 140–142, 252, 279, 293, 312, 321, 328–329, 332 mind vs. brain, 181 mnemonic devices, 230 monkeys and calculators/computers, 186 morphological change, 220 morphological plasticity, 221, 225 motor movement in mental computation, 88 motor-planning functions, 95 Müller-Lyer illusion, 38 mummification and treatment of the brain, 177 Mundurukú, xxiii, 13, 15, 77, 118, 151, 201, 206, 215, 339, 344–345 music incorporating material forms, 196 as the material control of auditory perception, 195 musical rasp, 270 mutually reinforcing forms, 227 Nadahup, 215 nativism, 32 nativist model, xxiii, 32–33, 50–51, 94, 116, 123 natural or counting numbers, 1 Neandertals, 60, 352 association with bone from Les Pradelles, 270 differences in creativity, 352 differences in material culture from Homo sapiens, 65, 272 differences in parietal lobes, 60 elongated brain shape, 61 evolutionary split with Homo sapiens, 122 hypothesis of indistinguishability, 64 smaller cerebellar volume, 64, 272 tool complexity, 65 necessary conditions, 116, 118–119, 121, 123 Nesher Ramla. See bone from Nesher Ramla

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  neural muscles, 90 neurocentric model of numerical cognition, 16 neurocentric view of numerical origins, 297 noncanonical expression of higher numbers, 300 non-neurocentric model, 16 nonsymbolic quantity, 48, 67 impervious to WEIRD/non-WEIRD divide, 38 North American calendar stick, 270 Nuer, 13 number defined in terms of properties, 3 essence, 3 formally defined, 3 working definition, 6 number systems, diachronic change, 40, 57 numbers always abstract, 344 always materially bound, 344 approximate or fuzzy, 13, 15, 100, 187 asymmetric understanding of, 12 as the basic elements of a mathematical system, 3 defined by their relations, 13 as equivalences or collections, 13 how old they are, 25 independent development in ancient languages, 111 as relational system, 334 relations, 4 scientific and nonscientific, 41 number-words, 98–100, 198, 293, 299, 303–304, 327, 329 oldest, 98 numeracy, xxii, 44, 50–51, 61, 126, 239, 332 participation by mobility-impaired individuals, 90 without language, 242 numerical discrimination, and graspability, 61, 214 numerical elaboration, 131, 149–150, 344 correlation with socio-material conditions, 126, 146, 171 five trends in the material chronology, 158 mechanism of, 149, 188 recruitment of new material forms, 88 retention of older forms, 88, 157, 159

selection of new material forms, 159 starting point for, 126 systematization in, 156, 159, 162, 221 numerical notations, 157, 172, 175, 182, 309, also see symbolic notations analyzed as a standalone technology, 311 change in nonsubitizable forms, 323, 326–327 ciphered forms, 182, 311, 323, 326–327, 333, 339 conservation of subitizable forms, 323, 327 properties of, 169 qualities of, 311 visual indistinguishability of elements, 46 numerosity, 43–45, 47, 50–51, 108, 121, 189, 322 in ancient peoples, 103 in hominoids, 50 in human infants, 49 multimodal, xviii neurological substrate, 59 the perception of quantity, not number, xviii phylogenetic distribution, 50, 121 in primates, 50 span of perceptual modalities, 68 in species ancestral to humans, 50 topographical structure, 67 visual dimension, xviii object tracking, 44 object-specified counting, 293 ochre from Blombos Cave, 237 octal, 21 offloading mental content, 42 Oksapmin, 15, 81, 98, 115, 207, 223, 226–227, 241, 244, 339 Old Akkadian period, 107 Old Babylonian period, 107, 279 one as number, 18 one-to-one correspondence, 91–92, 199, 221, 230, 239, 244, 255 in rosary, 92 ontogenetic maturation, role in numerical acquisition, 39 operations, 5 accumulating emerges first, 222 emergence of, 162

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  ordinal numbers, 47, 105 Mesopotamian, 106 ordinal sequences, 189 the counting numbers, 100 days of the week, 4, 63, 189 dedicated neural network for, 63, 87, 99, 189 letters of the alphabet, 4, 63, 100 ordinality, 4, 63, 105, 233 organic substances and archaeological survivability, 136 overdetermination, 257 pairing, 91 paleographic techniques, 266–267 panpsychism, 179, 185 parietal expansion, and tool use, 61 parietal lobe, 59–60, 63, 67 pebbles, properties of, 169 Penfield homunculus, 67 perceptual experience of quantity, 45, 70, 162, 173, 198, 203, 238, also see numerosity perceptual system for quantity, xviii, also see numerosity perishable materials, 239 persistence, 236 phonography, 326 phrenology, 178 Piaget, Jean, 30, 36–37, 39–40, 56, 111, 343 application of work to societies, 37 cognitive structures, 36 Piagetian developmental theory, 36 pineal gland, 178 Pirahã, xxiii, 118–120 Plato, xxv, 10, 25–26, 145, 351 plurality, the Greek concept of where numbers started, 18 Pomo, 138–139, 142, 245, 251–252, 260, 274, 315 positional number system, 284 positionality, 339 possible signs of forgetting, 338 pre-language or protolanguage, 122 primates, 54, 122 prime numbers, 11, 274, 304 procedural memory, 97 processes of linguistic change, 98 productive base, 6, 24 productive grouping, 14

Proto-Afro-Asiatic, 142 proto-cuneiform notations, 266, 314, 333 Proto-Elamite, 11, 328 Proto-Indo-European, 83, 101, 142 proto-languages, 41, 82, 100, 130, 141 Proto-Quechua, 137 Proto-Semitic, 103, 142 quaternary, 21 quinary, 21 quinoa, 282 Rapa Nui, 337–338 rational numbers, in chimpanzees, 50 reading/writing as extended/enacted cognition, 180–182 realism, xxv, 10–11, 25, 33 recognition of written characters, 174 recording European Middle Ages, 164 recursion, 119–120 in flintknapping and knotting, 121 in human evolution, 121 in mathematics, 119 in other species, 121 redrawing the boundaries of cognition, 115, 166, 177, 184 reference set, 4, 77, 239 regional pattern, 129, 132–133 reification, 56 relations, rearranging loose elements, 277 relative age, 94, 102–103, 128 representing vs. manipulating, 149, 163 reptiles, 50 restricted number systems, 134 Roman numerals, 11–12, 14, 18, 286, 319, 327, 329, 344 rosary, 239–240, 245 rosary, prehistoric beads possibly used as, 8 routine and habit, 88 rule-based production, 112, 114, 119, 121 Russell, Bertrand, 54, 344 definition of number, 3, 55 importance of magnitude ordering, 63 one-to-one correspondence, 91 theory of logical types, 54 salamanders, 49 Salamis tablet, 279 San, 269

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  script, 174 senary, 21 sensorimotor cortices, topographical layout of, 67 sequentiality, 187 Sessuto, 346, 348 sexagesimal, 21, 330 Sexagesimal System S, 103 shibboleths, 134 sign language, idiosyncratic, development of, 86 signification, 312, 322, 326 situated cognition, 179 Skhul Cave. See beads from Skhul Cave small, isolated societies, 100, 130, 132, 146 socially privileged numbers, 225 societal modes of thinking, 37, 343 socio-material conditions, xxiii, 99, 131–132, 137, 140, 143, 146, 171, 219, 250, 252, 258, 263, 274, 281, 350 difficulty in comparing in ancient societies, 146 difficulty in comparing in modern societies, 146 span of numerical elaboration, xx spatial-numerical association of response codes (SNARC), 58 subbase, 14, 21 subitizable subgroups, 199, 323, 327–328 subitization, 45 subitizing range, 214, 241 effects on material devices and language, 46 and first numbers to emerge, 46 subtraction in forming number-words, 300 succession, 233 successor function, 119, 308 sufficient conditions, 116, 118–119 Sumer, 14 Sumerian, 101–104, 106, 108–109, 112, 114, 142, 156, 216, 300, 328, 330 Sumerian corpus, 106 Sunghir, 138, 140, 142–143, 147, 219, 245, 252 supramarginal gyrus, 61 sustained, collaborative use, 173, 182, 193–194 symbolic notations, xxi, 38, 162, 173, also see numerical notations

symbolic quantity, 48 synesthesia, 63 synthesis, 56 Taï plaque. See bone from Taï tally, 336 in the Amazon, 136, 230 d’Errico’s encoding factors, 255 d’Errico’s six social purposes, 257 as device, 155, 163 differentiating unintentional and intentional marks, 191, 254, 259 difficulty of removing notches, 162, 222 discerning purpose of intentional marks, 255, 267 emergence in prehistory, 243 forms assumed to be, 246 Hayden’s identifying traits, 259 individuated or tokenized elements, 234–235, 239, 243 knowledge required to understand, 336, 338 one-to-one correspondence, 92 use of perishable materials, 136 place in material chronology, 148, 159 properties of, 149, 169–170 reasons for using, 172 representation marks, 259 stripes of mud, 92, 222, 240–241, 244, 254 tallying defined, 230 techniques, shortalls in, 254–257, 276 temporal accumulation, 255–256 tokenization, 235 tokenized elements, 232, 235 tokenized logic, 330 types and forms, 220–221 undifferentiated marks, 255, 258–259, 263 ungrouped tally, 207 unintentional marks, 225 verifying the number of notches, 228 visual indistinguishability of marks, 46 tally-counting, 279, 290, 292–293, 297, 338–339, 345 ceremonial and public, 296 Mangareva binary variant, 293 material record of, 292 physical and mental efficiencies, 292 remainders in, 291 tangibility, 149, 195, 197

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  target set, 77, 239 technical expertise, 65 techniques for interpreting possible tallies, 141 technological distance, 160–161, 170–171, 229, 231–232, 234, 240–241, 345 temporal lobe, 174, 324 terms for five and ten, cross-linguistic basis of, 76 text corpus, 104 differences between ancient and modern writing, 106 Thao, 96 thought experiment with children arranging or naming objects, 190 threshold of noticeability, 47 TM.75.G.2198, 329 tokenization, 233 through artistic forms, 236 and complex motifs, 237 through motor and visual regularities, 238 transfer between material forms, 237 tokens manipulable forms like coins, 172 properties of, 169 techniques, shortalls in, 266–267 tokens, Mesopotamian, 113, 141, 151, 156, 266, 279, 283, 304, 309, 312, 314–315, 333, 335–336, 343 organized linearly, 160 Tonga, 13, 293 tool use, evolutionary history, 123, 243 Trumaí, 75 Tswana, 214 Tukano, 75 two as number, 18 type and token, 223, 232, 234, 236, 241 unanalyzable forms, 84–85, 96, 128, 213 unbundled maximum, 252, 266, 327 uncontrastive system, 241, 244 the unity, 18 universality, 10, 145 Upper Rio Negro cultural area, 133, 135, 228, 230 Uruk V period, 103 using the body to measure space, 196 using the body to produce noise, 196

Venerable Bede, 208, 279 vigesimal, 21, 299 visual choice, 200 visual experience of the hands, importance of, 71 Visual Word Form Area, 174 wampum, 138–140 Warekena, 135, 230, 233 Wathaurung, 240, 244, 254 wayfinding, 272 Weber’s Law, 47 Weber–Fechner constant. See Weber’s Law WEIRD societies, 38, 118 WEIRD/non-WEIRD divide, 38 Western idea of number, 10 backward appropriation, 10–11, 287, 297 Western numbers, xix, 5, 13–15, 40, 57, 163, 226, 242, 284, 287, 302, 304, 308, 337, 342, 344 adopted by other societies, 9 ancient roots of, 7 backward appropriation, 338, 348 change over time, 17 contextualized, 127 differences from other cultural systems, 10, 13 productive base, 6 Western numerals, 311, 327 whole numbers, 3 Wittgenstein, Ludwig, 344 Wittgenstein’s ladder, 344 working memory, 44, 64, 326 writing ambiguity of signification, 326 automaticity, 326 balance in level of detail, 325 as being language, 320 as both noun and verb, 180 continuity of descent and contiguity of function, 309–311, 321 cursive as trade-off between ideal form and speed, 326 depictiveness, relaxation of, 324 differences between numbers and nonnumerical notations, 312 discrimination of characters, 324 individuation of characters, 324 as a material form, 180

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  writing (cont.) points of contrast, 324 shared function and form with precursors, 182, 310 standardization, 326 tolerance for ambiguity, 324 topological recognition of characters, 324 use in calculating, 335 writing systems, 104 differences between ancient and modern writing, 104 written notations, 338 fixedness in, 163 and lack of sound values for numbers, 157, 327–328 Xerénte, 215 Xerxes I, 340–341 Xhosa, 96, 214

Yamana, 206 Yanoama, 74 Yoruba, 298, 300–301, 303–304, 306, 308, 333, 343 Yoruba, ten different ways to express 19,669, 301 Yucuna, 215 Yuki, 14, 79 yupana, 135, 164, 281–284, 286, 296 use of perishable materials, 136 Yuruti, 230, 233 Zande, 209 zero, 14, 17, 19, 287 Babylonian roots of, 17 in honeybees, 50, 95 initial suspicion of, 18 as a number, 18 as metasign, 17

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