The question of where numbers come from is perhaps one of the last great mysteries of our time. Today, numbers are seemingly everywhere, and yet, they are nowhere to be found in nature. Ancient Greek philosophers like Plato thought that numbers were universals, eternal concepts that were in a sense real, albeit differing substantially in the nature of their existence from the stuff of the physical world. Certainly, all societies that count form the exact same sequence (“one, two, three…”). That sequence contains the exact same patterns, as for example, the primes, numbers that can only be divided by themselves and one. These qualities indeed give numbers the air of existing in some way external to the human mind, so they do seem to be “out there” somewhere waiting to be discovered.
Within past decades, mathematicians and cognitive scientists have positioned numbers as mental creations, introspected and conceived by the large and powerful human brain. Significantly, the human species shares with other animals an ability to appreciate quantity, known as numerosity. Researchers who study this ability often aver that it alone is sufficient for numerical thinking. The problem with this perspective is that while all humans have this so-called number sense, not all societies have numbers. Some societies count no higher than two or three, and one society—the Pirahã of Amazonian Brazil—is credited with having no numbers at all. This circumstance suggests that developing the kinds of number systems found across the planet requires something beyond numerosity. Linguists suggest this extra ingredient is language. The problem with this perspective is very similar: While all human societies have language, not all of them have numbers. So, if numerosity alone or numerosity plus language cannot satisfactorily explain numerical origins or cross-cultural numerical variability, something else is still needed.
Archaeology offers a plausible answer. Why archaeology, of all things? This is a reasonable question, since numbers are not the sort of object that can be dug up from the ground or analyzed in the lab. However, as the science of material culture, archaeology has a unique perspective on numbers: their material component. These are the devices used to represent and manipulate numbers—things like fingers, tallies, counters, and written notations. These become important if we reverse the usual order of things. Rather than thinking of material forms as the passive recipients of our externalized mental content, we instead recognize our manuovisual engagement of material forms as a mechanism for generating ideas about numbers. For example, if we count on our fingers to five, as many societies do, we get one numerical structure. If instead we count the spaces between our fingers, as the Yuki of Northern California once did, we get a completely different structure. Instead of decimal, our numbers will be based on four or eight, depending on whether we use one or both hands.
The use of material forms not only connects the ability to appreciate quantity to the development of symbolic notations, it can also explain why different societies have different numbers, including their extents and organizations. Symbolic forms and associated concepts of number emerge from the perceptual experience of quantity through a sequence of material forms systematized by our capacities and their affordances, starting with the hand (Figure 1). Over time, the symbols and concepts become increasingly different from the original percepts: They acquire properties from the material forms used to represent them. Their explicit forms become increasingly concise and involve a larger component of implicit knowledge that the user must supply. They become distributed over multiple forms, which makes them independent of any particular form. They also change from equivalences (“as many as the fingers on one hand”) to collections (five discrete notches or pebbles) to entities (five as an object in its own right). Concepts and properties transfer across the different devices in the form of habits, expectations, and knowledge. Cross-cultural variability becomes a straightforward matter of whether a society uses material devices for numbers, which specific devices it uses, and how it uses them.
The sequence of material devices is also systematized by what devices can and cannot do. As representations of number, the fingers lack capacity and persistence. When a society needs both, it will recruit a device like a tally that can do what the fingers do—accumulate and represent—while adding the required capacity and persistence. While adding capabilities, the notches on a tally also bring in new problems: They are difficult to appreciate whenever they number more than about three or four, and they cannot be moved or removed once made. This will motivate the society to recruit loose objects as counters, typically things like pebbles or kernels of corn. Counters add manipulability. However, they are unsuitable for representing, so written forms might be added. Viewing numbers as concepts realized and elaborated through the use of material forms is empirically verifiable through the many correspondences between the properties of devices used and the conceptual properties of numbers.
An archaeological perspective on numbers requires some flexibility on what constitutes a “material” form, as fingers and notation are not solid physical substances in the way that tallies and abacuses are. Instead, fingers are part of the body, biological and alive. They do not readily preserve, even as fossils, and when they do, they obviously do not represent gestures like finger-counting. Even the behavioral traces preserved in the archaeological record do not let us recover such ephemeral gestures. But contemporary linguistic and ethnographic evidence shows that the fingers and sometimes the toes are used when numbers are emerging, so their structure—five digits per limb—influences numbers toward patterning by fives, tens, and twenties. The visual experience of the hand also influences numbers toward discreteness, especially about three or four, the upper limit of what the number sense appreciates. The hands also help us realize five and ten, the first discrete quantities above that range.
Gesture helps us express things for which we lack words, and which we might not even clearly understand ourselves. Using the hand to express and count numbers is unique to humans. Chimpanzees do not express numbers gesturally or count on their fingers, even though their hands, brains, and behaviors are closely related to ours. People who are congenitally blind do not express numbers gesturally or finger-count either. This suggests that using the hand to express and count numbers is not simply a matter of neurological predisposition or language. Rather, the hand is a material object that we see and engage manuovisually. This is distinct from using the hand to sense or act, where seeing it is not as essential. And because the hand is used as a material device for counting, it becomes relevant as such to an archaeological inquiry.
As for written notations, we tend not to think of them as material because we focus on their symbolic quality. We also become sidetracked by the surfaces we write on and whatever substances and tools we use to make the marks. However, if we consider the characters themselves as material forms, we see that—drawing on attributes added by tallies and counters—early numerical notations represent quantity by repeating elements and arranging them in small groups with appreciable quantity, typically threes and fours. Over time, symbolic numbers emerged as these written characters became recognized through their physical features and spatial relations and were written more efficiently and quickly. These developments meant that numerical signs no longer needed to represent their meaning through element quantity, but could simplify as conventions. The process itself was simple: Each generation merely used the signs in their present form, adapted behaviorally and neurologically to their use, modified the signs just a bit, and passed them along. And by changing in response to use, the material signs acted to accumulate and distribute changes in behaviors and brains, until character forms could no longer be recognized and used without training and practice.
By considering material forms—not just tallies and counters, but also fingers and written notations—as having an active role in the emergence and elaboration of numbers, we become able to explain how symbolic notations emerge from the perceptual experience of quantity, why different societies have different numbers, and why numbers are so universal. Numbers are, after all, the product of species-universal resources and systematized material forms, so it is no coincidence when they produce strikingly similar outcomes across significant differences of time, place, and culture. We also open the door to considering materiality as a component of human cognition more generally, along with new possibilities for understanding what human cognition is and what roles material forms might have in it. This inquiry into the origin of numbers and the very nature of the human mind is an exciting challenge, one that the science of material culture is uniquely equipped to undertake.
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