Introduction: Three reckonings

This is a book of three reckonings.

Firstly, to reckon is to calculate. In the modern world, calculating, working, and playing with numbers are far more important activities than in any earlier period. However, numeracy (the quantitative counterpart of literacy) is fundamental to the functioning of all large-scale societies, including premodern ones. It is central to trade, the appropriation of wealth, and the redistribution of surpluses. And working with numbers, at a smaller scale, is part of all sorts of everyday practices: measuring out quantities for recipes, working out the days until the next season of a favorite show is released, or tallying the number of guests at a party. Reckoning is a (nearly) universal social practice that has taken on new meanings and new functions in capitalist and industrialized societies. The mathematician John Allen Paulos (1988) correctly diagnoses, though, that despite the ubiquitous nature of arithmetical problems in modern, industrial lives, people don’t always have the tools needed to work adequately with numbers. Sometimes this can create enormous feelings of inadequacy. As a psychological response, some people take a perverse pride in their innumeracy, in a way that we would never do with illiteracy. In contrast, I admit to taking some perverse interest in the historical and social scientific study of mathematics and numbers, a subject that many of my colleagues who are anthropologists find baffling or even disgusting. (Some of them even got into the discipline to avoid taking more math!)

Reckoning—counting and calculating—keeps a suitable semantic distance from words like mathematics. This book is not a history of mathematics, and you do not need any training in the field to read it. Understanding how people, both past and present, reckon—how they work with numbers as part of daily life—helps us grapple with their lifeways. Those of us who learned arithmetic using pen and paper, working with the ten digits 0–9 and place value, may find it natural and take for granted that this is the way it’s always been done, or at least the way it ought to be done. But think of the amount of time and energy spent in the early school years just to teach place value, and you’ll realize that this sort of numeracy is not preordained. And that’s not even considering that pen-and-paper arithmetic requires the widespread availability of cheap writing instruments and media as seemingly simple as paper, which shouldn’t be taken for granted.

Over the past 5,500 years, more than 100 structurally distinct ways of writing numbers have been developed and used by numerate societies (Chrisomalis 2010). Thousands more ways of speaking numbers, manipulating physical objects, and using human bodies to enumerate are known to exist, or to have existed. Each of us is familiar with only a tiny fraction of the diversity of the world’s number systems. The current universality of a particular set of these practices among Western, Educated, Industrial, Rich, and Democratic (or WEIRD) societies leaves much of the human condition uninvestigated (Henrich, Heine, and Norenzayan 2010). Because this diversity in reckoning practices has only partially been described, we need a better understanding of how people work with numbers. In this book, I draw on, and expand upon, the enormous cross-cultural and comparative literatures in linguistics, cognitive anthropology, and the history of science that bear on questions of numeracy.

Secondly, to reckon is to think. The etymological and semantic linkage between thinking and calculation is strong in English, as it is throughout large swaths of the Indo-European family of languages. We draw semantically on the taxing mental activity of computation, extending it metaphorically to thinking in general. A reckoning is an estimation or judgment that is undergirded by the mental work necessary to reach it. Explaining the history of number systems relies on understanding the mental, verbal, and symbolic manipulations that mathematical cognition requires. These cognitive operations are embodied through language and gesture, and they are materialized through artifacts and notations (Overmann 2016). Because numeracy is not just a social process but also a cognitive one, this is a book about reckoning about reckoning.

This is not a formal cognitive analysis, however. I present no technical accounting of how past people thought while manipulating numbers. Such an analysis is surely impossible for any period but the present. Where relevant, it draws on experimental cognitive psychological analyses of numerical cognition, some of which are informed by anthropological insights (Rips 2011; Carey 2009). But often, historical and cognitive disciplines have worked in parallel rather than in tandem. The anthropologists, historians, and linguists who work on numeracy and numeration must be aware of these important cognitive scientists. These include my fellow cognitive anthropologists, many of whom have undertaken vital empirical studies of numeracy and mathematics in the daily lives of people (Hutchins 1995; Mukhopadhyay 2004; Marchand 2018). In many cases, however, because this is a book centered on past cognition, I draw much more substantially on the work of cognitively informed humanists such as the anthropologist Jack Goody (1977), the historian of science Geoffrey Lloyd (2007), and the classicist Jocelyn Penny Small (1997). These analyses use fragments of historical, linguistic, and archaeological evidence to examine human cognition across long periods, in light of the impossibility of direct experiment or observation. Within anthropology, cognitive archaeology is the subfield most directly concerned with historical processes of numerical cognition, and I certainly draw on that work as well, where relevant (Morley and Renfrew 2010). I also draw on the considerable and growing body of material on numerical cognition in small-scale societies from cognitive anthropology and cross-cultural psychology, which asks ethnographically informed cognitive questions about the lexical, embodied, and material numeracy of groups traditionally disparaged or ignored in histories of mathematics (Saxe 2012; Everett 2017).

Finally, building on these first two reckonings, to reckon is to evaluate, to assess, and to assign worth. This third reckoning is imbued with conflict and tension—as when we are forced to reckon with some new circumstance. We make choices and judgments that have consequences. When it comes to number systems, the prototypical instance of this tension is the replacement of the Roman numerals (for most purposes) by the Indo-Arabic or, better named, the Western numerals. The case study of this replacement forms a thread throughout this book, although it is just one strand of a much broader tapestry. The traditional narrative holds that late medieval and early modern people, evaluating these two systems, chose the more efficient one and abandoned the more cumbersome Roman numerals.

Here I set out a different approach, one that draws on the literature on the disappearance and abandonment of writing systems (Baines, Bennet, and Houston 2011). I acknowledge that people make decisions, and that those decisions have reasons that make sense (both to the decision makers and, with a little work, to us too). If that were not possible, if every “other” were hopelessly foreign to our minds, a cognitive and comparative analysis of numeracy would not be possible. Understanding how people’s decision-making processes work, not just at an individual scale but at a collective and social one, helps us understand why one set of practices might not persist. But the reasoning we might use to make those judgments need not be universal—in fact, universal principles of rationality are very unlikely to provide full and satisfactory explanations. Figuring out how and why people abandon one way of doing things in favor of another is not trivial. It is unlikely to be subsumed under a small number of cultural “laws,” but it is also too important for us to simply throw up our hands and claim, in a particularistic fashion, that there is no predicting or explaining things.

I recall quite clearly, in the way one recalls times of extreme stress, a question posed to me at my PhD defense in 2003, by the pro-dean of my doctoral committee. This person’s job was to read my work from a completely different discipline from mine (in this case, biology), having had no prior contact with me, to ensure that everything was on the up and up, and that they weren’t giving away degrees to just anyone. During the defense, this biologist queried whether numerical notations should be conceived as analogues of organisms: after all, they are born, they have their natural life cycle, and then they die. My response at the time, which I still think is right, is that a better analogue is the species. While an organism has a limited and relatively inflexible life course, species evolve (out of ancestral ones), change over time, persist for an indeterminate period, and survive and thrive (or not) in different contexts; when they eventually are replaced or go extinct, it is because they are not well suited for the environment they then find themselves in. Just because one numerical system gives rise to a descendant does not mean that the ancestor must go extinct, and a system that is ideal in one context may be unsuited for another. We must take account of these longer-term, larger-scale historical processes as well as the local and particular.

Of course, the scale of biological evolution and the processes of cultural evolution are quite different, and we must be careful not to ignore the human element in invoking processes like “selection” and “adaptation.” One of the most serious and most reasonable charges against cultural evolutionary studies of cultural transmission is that they are insufficiently attentive to the contexts in which decisions to adopt, reject, or transform are made. So here, in dealing with the reckonings and evaluations of users about which numeral systems to prefer, we must be careful not to assume a universal logic that requires a particular outcome. Because people are not rational choice machines, we need to understand their cultural rationality in order to understand how they made sense of their decisions. This is particularly important when looking at people far removed, chronologically or culturally, from contexts with which we are familiar. Nonetheless, this is an evolutionary book, in a historical sense, seeking to understand the long-term processes by which the reckonings of individuals at discrete periods come together to shape the histories of numerical systems, until they too, inevitably, meet their day of reckoning. Not only are we not at the “end of history” of numeration—surely new systems will be developed and used in the coming centuries—but there can be no such end, as long as humans are still judging and evaluating their numerical tools.

Numerals, cognition, and history: these three reckonings are the three central themes of the following seven chapters. Each element of this triad is essential for understanding human numbering practices and the social context of written numbers. Or so I reckon.

Note: On “Western numerals”

Throughout this book, I use the term “Western numerals” to refer to the set of signs 0123456789, organized in a base-10 system using place value, as I have done in my earlier research. In the English-speaking world, we mostly learn these signs under the name “Arabic numerals,” which reflects the fact that they were borrowed by Western Europeans from Arabs living in Spain, Sicily, and North Africa in the tenth century CE. In the scholarly literature on numerals, these are most often called “Hindu-Arabic numerals,” which reflects a little more of the history of the system, because the Arabic script got its numerals from an antecedent system used in northern India as early as the fifth or sixth century CE. Other terms like “Indian” and “Indo-Arabic” are also found. The historian of mathematics Carl Boyer, whose early work on numeral systems played an important role in my development as a “numbers guy,” argued somewhat facetiously that we might more properly call it the “Babylonian-Egyptian-Greek-Hindu-Arabic” system (1944: 168)—although in this case I think he was wrong, and that “Egyptian-Mauryan-Indo-Arabic” would get the history straight.

The most basic problem with the formulations “Arabic” and “Hindu-Arabic” is that they do not adequately distinguish the set of signs 0123456789 from the set of signs used in Arabic script or the set of signs used in the modern Devanagari script, or any number of other decimal, place value systems. All of these descend ultimately from that same fifth–sixth century CE Indian ancestor. To make matters more confusing, in Arabic the numerals used alongside Arabic script are called arqam hindiyyah (Indian numerals). The problem of ambiguity is thus a serious one. Because several such systems are in active use (particularly the Western European 0–9 and the “Arabic” set), it becomes a nightmare to try to distinguish these systems from one another. We need different terms for each set of numerals.

Structurally they are very similar to one another—although not completely; for instance, many Indian writers customarily write 100,000 as 1,00,000 and 1,000,000 as 10,00,000. So I talk about Western, Arabic, and Indian numerals to refer to the decimal, place value systems used in three different script traditions. Paleographically—in terms of the history of the signs themselves—they are quite distinct, and are likely to remain so. One could argue that just as we talk about the “Latin alphabet,” we could call 0123456789 the “Latin numerals” instead of “Western.” But this would only create confusion with the “Roman numerals.” “Western numerals” reflects the fact that the particular graphemes (the specific signs) were developed in a Western European context and were first and most prominently used in Western Europe.

One might argue that by calling them “Western numerals” I am denying them their history, obscuring the fact that they derived from Indian and Arabic notations, which I certainly do not wish to do. But I think that Boyer has a point—why stop at “Hindu,” since the “Hindu” place value numerals derive from a nonpositional system used in Brahmi inscriptions in India as early as the fourth century BCE, which in turn probably derives from Egyptian hieratic writing going back as early as the twenty-sixth century BCE! And if we later decide that this history is wrong, do we then change the name? I am far more concerned that by using terms like “Arabic” or “Hindu-Arabic” for 0123456789, we render invisible the continued existence and active use of actual Arabic and Indian numerals in the modern Middle East and South Asia. Using an umbrella term—which, in reality, obscures all but a single variant of a rich family of numerical forms—unfairly collapses this complex genealogy with several extant modern branches into a single unilinear history. The history of place value becomes merely our history of place value. And, in the same way that the fallacious evolutionary error that humans are descended from chimpanzees renders chimps as our ancestors when they are actually our cousins, we must avoid rhetoric that suggests that Devanagari, Arabic, Persian, Telugu, Gujarati, and many other decimal positional systems are historical relics. No one seriously disputes the facts of the history and evolution of these systems, but our labeling practices run the risk of making it appear as if we stand alone at the end point of the history of numerals. “Western numerals” highlights that specific paleographic and structural innovations happened in the West (principally in Spain and Italy), but maintains a suitable conceptual distance from the related but still vital systems of the Middle East and South Asia.