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.

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

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 cross-culturally.

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.

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.

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.

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.

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.

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.

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.

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?