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?

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.

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.

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.

Recent advances in the field of palaeogenomics have revealed that at the onset of the Late Neolithic, Europe was characterized by a major cultural and genetic transformation triggered by multiple population movements from the Pontic–Caspian steppe. Corded Ware populations show a large-scale introduction of Yamnaya steppe ancestry across the entire archaeological horizon (Allentoft et al. 2015; Haak et al. 2015; Malmström 2019). The emergence of the Bell Beaker burial identity in the early third millennium BCE was similarly accompanied by a dramatic genetic turnover, at least in Northwestern Europe (Olalde et al. 2018). These population changes call for the integration of genetic evidence into existing models for the linguistic Indo-Europeanization of Europe (cf. Kristiansen et al. 2017).