C H A P T E R 6 .1 The cognitive and cultural foundations of numbers Stephen Chrisomalis E very known human society has the capacity to manipulate quantity; in this sense, mathematics is panhuman. The extent to which any individual or group pays special attention to mathematical concepts, however, is crossculturally variable. The study of the cultural, (pre)historical, and social aspects of numeration and arithmetic—the foundations of mathematics—is preoccupied with the tension between the universal and specific aspects of the subject. While recent scholarship from cognitive science, generative linguistics, and neuropsychology provides ample evidence that there are some ‘hard-wired’ aspects of human numerical faculties, developmental psychologists, ethnomathematicians, anthropologists, historians, and historical linguists stress the differences among cultures and historical periods. How, then, can we synthesize and reconcile these disparate literatures? To assume, a priori, either that universal aspects of numeration or local and historically contingent developments are of primary interest is premature and misguided. To add to the confusion, the history of numeration lingers under a series of pervasive myths. Nineteenth-century unilinear notions of the evolution of society from ‘savagery’ or ‘primitiveness’ to ‘civilization’ continue to haunt the subject. Such notions have persisted despite a paucity of evidence, partly because the data to refute them are spread among multiple disciplines, and partly because 496 people and practices there is a sufficient kernel of truth to them to warrant further investigation. A comparative and historical perspective can help debunk the untenable and shore up the sustainable propositions. Every known human language has some means of expressing quantity, and in virtually all languages this includes two or more words for specific integers (oral only in non-literate societies, oral or written in literate ones). Most societies, perhaps all, possess one or more means of conducting arithmetic mentally or with the aid of various artifacts and techniques that serve as computational technologies. Finally, a smaller number of societies have a set of visual but primarily nonphonetic numeral-symbols that are organized into a numerical notation system. Western numerals and Roman numerals are merely two well-known examples among the many distinct systems used over the past 5,500 years.1 The linkages between number words, computational technologies, and number symbols are complex, and understanding the functions each serves (and does not serve) will help illustrate the range of variability among the cognitive and social systems underlying all mathematics. Number words and number concepts Although linguists, particularly European philologists, have been interested in numeral systems for centuries (Pott 1847; Kluge 1937–42; Salzmann 1950), most of this work was largely non-theoretical description of the numeral systems of the world’s languages. In the 1960s, both the generative grammar framework pioneered by Noam Chomsky and the research into cross-linguistic universals led by Joseph Greenberg used numerical evidence as support for panhuman linguistic capabilities, leading to a renewed interest among psychologists, philosophers, and linguists in the foundations of numerical systems in the human language faculty. Linguists are divided about just what the existence of a universal number concept means in terms of human evolution and the existence of an ‘innate’ number sense. Chomsky (1980, 38–39) believes that natural selection could not have selected for the human number concept, and thus contends that it is qualitatively different from the quantificational abilities of apes and other animals. Mathematics must, therefore, be no more than a by-product of some other evolved ability, such as the language faculty. For Chomsky, the language faculty and the number faculty (and these two faculties alone) share the concept 1. I use the term ‘western’ to refer to the signs 0123456789 instead of ‘Arabic’ and ‘Hindu-Arabic’, not to deny that this innovation was borrowed from a Hindu antecedent through an Arabic intermediary, but to avoid confusion with the distinct Indian and Arabic numerical notations used widely to this day. Rendering these latter notations ‘invisible’ through nomenclature is counterproductive and potentially ethnocentric. The cognitive and cultural foundations of numbers of ‘discrete infinity’—the ability to create an infinite number of things from a smaller and finite set of discrete symbols (Chomsky 1988, 167–168). However, while all languages can, in theory, produce an infinite number of sentences or phrases from their existing vocabulary, there is no natural language whose number sequence can be extended indefinitely without the creation of new numeral terms (Greenberg 1978, 253). For instance, English dictionaries normally end the number sequence at ‘decillion’, and one needs a new number word to form ‘one thousand decillion’ in a regular fashion. This is in direct contrast with place-value numeral notation systems, in which one may add zeroes to a number ad infinitum. Unless one wishes to argue that the number concept is divorced entirely from the numeral words, this is poor evidence that the number concept derives from the language faculty. Nevertheless, attested languages display considerable regularities in the structure of their numerical systems, which suggests that some underlying principles severely constrain or even determine the range of outcomes (Hurford 1975; Greenberg 1978; Corbett 1978). A more sophisticated version of this hypothesis is that promoted by the cognitive linguist Heike Wiese (2003; 2007). Wiese argues, based both on her empirical research and on the mathematical philosophy of Frege (1884), Russell (1903), and others, that numeral words ‘do not refer to numbers, they serve as numbers’ (Wiese 2003, 5). The number concept is a byproduct of the language faculty; numerals developed once language did, as a means of labeling objects within the context of counting activities (for example, naming the fingers, or pebbles, or other physical objects), and thus became the numbers themselves. It is common cross-linguistically for the names of numerals, particularly ‘five’ and ‘ten’, to be connected etymologically to the fingers or hands (Bengtson 1987). Hurford (1987) argues along similar lines that one possible origin of number words is through counting rhymes and games such as ‘eeny-meeny-miney-mo’, rather than as an automatic linguistic expression of an underlying concept. This interdependent co-evolution of numeral words and numerical concepts is a reasonable proposition and is congruent with much of the linguistic literature on the subject of number, but as a historical or evolutionary hypothesis remains untested at present. In contrast to these perspectives, which stress the universality of the number sense but also deny that it evolved specifically under natural selection, Stanislas Dehaene (1997) and Brian Butterworth (1999), working from neurological and psychological foundations, argue that the structure of the brain will tell us a great deal about how humans count and compute. One line of evidence suggesting that this is so is that various animals have been demonstrated to possess quantificational abilities, particularly the ability to distinguish small quantities up to three or four (McComb et al 1994; Hauser 2005). Pre-cultural and pre-linguistic infants have been shown to possess such abilities as well, by means of attention 497 498 people and practices studies that examine the attention and gaze of infants at artificially unexpected or counterintuitive numerical situations (Wynn 1992; 1998). Regardless of language, humans can more easily distinguish four from five objects in a group than nineteen from twenty objects, and more easily distinguish eight from twelve objects than nine from eleven—suggesting some sort of intuitive, plausibly hardwired analog numerical representations. Yet, as Carey (2001) points out, even if this is the case, the number line itself, based on the successor function of discrete integers, may nonetheless be a cultural construction and may not be evolutionarily hardwired. Geary (1995) and Miller and Paredes (1996) have discussed the differences between Chinese and English numerical systems and their cognitive consequences in terms of ease of learning for children, suggesting that even where there are universal aspects of numeration, variability also plays a major role. While the ability to distinguish two lions from three seems relevant from the perspective of natural selection within the evolutionary history of hominids, the ability to do most arithmetic, or to organize numerical systems in terms of a base and its powers, does not. If these are indeed universal phenomena, other explanations are needed. Culture, number, and cognition From early in the study of other societies, the absence or relative paucity of numeral words has been regarded as evidence of savagery among the indigenous peoples of the world. Perhaps the most notable statement of this sentiment is ‘On the Numerals as Evidence of the Progress of Civilization’ by the Scottish surgeon-scholar John Crawfurd, then president of the Ethnological Society (Crawfurd 1863). Crawfurd’s position was that numerals were among the last words invented in any language, and that they ‘advance with the progress of civilization’, and thus that the ‘social condition of a people is, therefore, in a good measure, indicated by its numeral system’ (Crawfurd 1863, 84). Nothing about the concept of natural selection implies progress. Nevertheless, following the publication of Charles Darwin’s Origin of species in 1859, Darwin and his colleagues, working in a Victorian context where British imperial rule was nearly unchallenged, often wrote and behaved as if culture evolved in a single line from simple to complex. In this social and intellectual context, which owed much to Enlightenment speculative histories and racialism, numerals could serve as an easily quantifiable surrogate for measuring cultural progress. Darwin’s cousin and friend, Sir Francis Galton reported of the Damara of Namibia that: they certainly use no numeral greater than three. When they wish to express four, they take to their fingers, which are to them as formidable instruments of calculation as a sliding-rule is to an English schoolboy. They puzzle very much after five, because no The cognitive and cultural foundations of numbers spare hand remains to grasp and secure the fingers that are required for units. (Galton 1853, 133) Galton’s report need not be taken at face value; he was not fluent in Damara, which in fact has more numerals than he suggests. His unflattering account was used by Conant (1896) and others as evidence for the proposition that small-scale societies were generally numerically incompetent. The notion that the inventory of numeral words in a language is evidence for or against its speakers’ degree of civilization remained current throughout much of the twentieth century. It is found prominently in the psychologically-informed ethnology of Lucien LévyBruhl (1966 [1910]), who distinguished ‘primitive’ numeration based on ‘configurations’ of small quantities (pair, triad, etc.) from true cardination and the successor principle. Yet the lack of any meaningful definition of ‘primitive’ and ‘civilized’ independent of this assertion renders these conclusions invalid; they do no more than assert that societies that lack extensive series of numeral words are representative of an earlier stage of human development. It nonetheless cannot be denied that there is some correlation between the size of the set of numeral words in a language and other aspects of social life. Greenberg (1978), in his study of universals of numeration, pointed out that all the languages with limited sets of numeral words were small-scale societies, and suggested the need to inquire further into the implications of this fi nding for cultural evolutionary studies. Divale’s (1999) study of two samples of sixty-nine and one hundred and thirty-six societies and their numeral-words revealed a reasonably strong correlation between the highest number normally expressible in a language, and the degree to which the speakers of that language relied on the storage of foods (especially grain cereals) to prevent starvation due to climatic instability. He reasons that one potential explanation is that societies need to quantify such foods in order to collect and distribute them (see also Steensberg 1989). The degree to which a society is able to marshal resources to store food on a large scale, in turn, is related to social size and complexity. This correlation is interesting and deserves further study, but is unlikely to be the sole or even the primary determinant of the size of the numerical lexicon. A recent controversy involves the ethnographic study of the Pirahã by the linguist Daniel Everett, who asserts that this group of Amazonian horticulturalists possess no numeral words whatsoever, as part of a general cultural constraint against referring to objects and concepts outside of immediate experience (Everett 2005). This represents the most forceful and best-documented instance of such an assertion, and comes from a lengthy period of ethnography over some decades. Although the Pirahã have had centuries of contact with Brazilians of Portuguese descent, including trade relations, Everett has never heard them use any numeralwords, although he recognizes that certain grammatical constructions have a 499 500 people and practices ‘quantificational smell’ (Everett 2005, 625). The Pirahã themselves are acutely aware of this absence, and expressed concern to Everett that their lack of arithmetic was hindering their trade relations with Portuguese-speaking itinerant traders, but nevertheless had limited success in an educational program designed by Everett to teach them quantification. Gordon (2004), based on a short field period among the Pirahã, presents some evidence from psychological testing that seems to suggest that the Pirahã lack of quantificational words extends to the perceptual and cognitive domains as well. Dixon (1980, 107–108), similarly, has argued that there are no true numeral words in some Australian languages. Everett’s assertion that a Pirahã cultural constraint inhibits their use of quantification and structures their thinking about the world is contentious but deserving of further study. Aside from the work of Everett and Gordon, there is some evidence that speakers of languages that have limited sets of numeral words also have specific limitations in numerical cognition. Findings from developmental psychologists such as Piaget (1952) and Vygotsky (1962) provide an independent set of criteria on which numerical cognition can be judged cross-culturally. Lancy (1983) undertook detailed psychological testing of members of various groups in Papua New Guinea and found that monolingual children who spoke local languages and had no formal education had considerable difficulty with tasks considered simple for their age. This work is supported by the massive linguistic research of Lean (1991). Yet as Gay and Cole (1967) note, traditional mathematical practices can in fact have cognitive advantages over those achieved through Western-style education; the Kpelle of Liberia, among whom they worked, could, for instance, more accurately estimate the volume of a pile of rice than Western-educated individuals, although they performed arithmetical calculations more poorly (see also Reed and Lave 1979). We must be cautious before inferring causation from correlation in these cases, and be wary of ethnocentrically projecting Western interests and values onto tribal societies. The presence or absence of many numeral words in these languages must be conceived in terms of perceived social needs (or lack thereof) rather than as an intellectual failing. Hallpike (1979, 237), who generally follows Lévy-Bruhl in asserting that abstract number concepts are absent from primitive societies, nonetheless cautions that numerical abstraction ‘cannot be deduced merely from the existence of a series of verbal numerals, even a series extending to 100 or 1,000 or more’. Hallpike stresses instead that only the presence of the right sort of social problems leads to the cultural evolution of formal-logical reasoning about quantity. There is abundant evidence that when numeral words are desired, speakers of any language are capable of extending their numeral word sequence, either through modeling new words on older ones in their own language, or through borrowing words from other languages. It cannot be ruled out that when the need is no longer present, higher numeral words cease to be used. The cognitive and cultural foundations of numbers Some languages use different sets of numeral words for counting different classes of object, or numeral classifiers. Some linguists and ethnographers argue that numeral classifiers represent evidence of ‘concrete’ counting as opposed to ‘abstract’ numeration. While in some cases this variability simply amounts to the use of different morphemes at the end of a single set of numeral words, in other cases the numeral words are radically different. Conant (1896) presented a group of numeral systems from the Tsimshian language of northern British Columbia, Canada (Table 6.1.1). Conant held that the use of multiple systems represented linguistic ‘primitivity’ and suggested a lack of numerical abstraction, and through Lévy-Bruhl and others this idea enjoys some currency in the contemporary study of numerals. The work of the Near Eastern archaeologist Denise Schmandt-Besserat (1984; 1992) on token systems of the prehistoric Middle East relies heavily on the notion that the use of different symbols (lexical or graphic) for the same numerical referent has cognitive implications for the users of such semiotic systems. This theory has been developed more thoroughly by Peter Damerow (1996), who notes that the multiplicity and semantic ambiguity of the numerical notation systems of late fourth-millennium Mesopotamia suggest an incompletely abstract number concept. Yet the languages that have numeral classifiers include the Maya languages, whose users developed complex astronomy, mathematics, and architecture (Berlin 1968; Macri 2000) and Japanese (Downing 1996), whose speakers can hardly be accused of non-abstract mathematical thought. In fact, numeral classifiers are no more than a taxonomic system akin to (though more specific than) grammatical gender. They may well reflect particular cultural perspectives on the classification of reality, but they do not imply that their speakers thus have no sense that gy’ap and kpal have an underlying ‘tenness’ any more than English Table 6.1.1 Tsimshian numerals with classifiers (Conant 1896, 87) No. Counting Flat objects Round objects Men Long objects Canoes Measures 1 2 3 4 5 6 7 8 9 10 gyak t’epqat guant tqalpq kctōnc k’alt t’epqalt guandalt kctemac gy’ap g’erel goupel gutle tqalpq kctōnc k’alt t’epqalt yuktalt kctemac kpēel k’al t’epqadal gulal tqalpqdal kcenecal k’aldal t’epqaldal yuktleadal kctemacal kpal k’awutskan gaopskan galtskan tqaapskan k’etoentskan k’aoltskan t’epqaltskan ek’tlaedskan kctemaetskan kpēetskan k’amaet g’alpēeltk galtskantk tqalpqsk kctōonsk k’altk t’epqaltk yuktaltk kctemack gy’apsk k’al gulbel guleont tqalpqalont kctonsilont k’aldelont t’epqaldelont yuktaldelont kctemasilont kpeont gak t’epqat guant tqalpq kctōnc k’alt t’epqalt yuktalt kctemac gy’ap 501 502 people and practices speakers are confused between six eggs and a half-dozen. With regard to late fourth-millennium Mesopotamian tokens and numerals, the accountants and scribes who used them were able to manage complex administrative tasks, and it is implausible that they did not recognize that ‘8 sheep’ and ‘8 bushels of grain’ had something in common. There is simply no evidence from existing human languages for Bertrand Russell’s assertion that, ‘It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2’ (Russell 1919, 3). Since the 1980s, the development of ethnomathematics has stressed that numerical concepts develop in different cultures in different ways, and that we should not dismiss too readily the achievements of non-western societies (Ascher 1991; Powell and Frankenstein 1997). Ethnomathematics provides a useful antidote to the sometimes aggressive Eurocentrism of earlier decades, and has brought contemporary anthropological insights to the study of mathematics, but balances a fine line between universalism and radical relativism. It is difficult to know what to make of Mimica’s claim (1988) that the Iqwaye of Papua New Guinea developed the concept of transfinite numbers on the basis that an informant used the same numeral word for ‘one’, ‘twenty’, and ‘four hundred’ (that is, x = x2 = x3). Similarly, despite Urton’s fascinating assertion (1997) that the Quechua number concept is strikingly different from the western standard, and thus that the ontology of numbers is culturally relative, it is extremely difficult to evaluate such statements in the absence of some criteria for evaluating the universality of the concept in the first place. Crump’s (1990) anthropologically and psychologicallyinformed synthesis remains the best of this work to date, forcing us to recognize both similarities and differences in number concepts. Ethnomathematics highlights the fact that there can be differences in numerical cognition that do not imply necessary distinctions between right/wrong, simple/complex, or primitive/ evolved. Tallying and abacus methods Alongside the universal or nearly universal employment of numeral words, the use of notched sticks, knotted strings, and other artifacts for recording number is similarly quite widespread cross-culturally. This has been thoroughly demonstrated for African societies (Lagercrantz 1968; 1970; 1973; Zaslavsky 1973), and more sporadically elsewhere in the world. Yet some objects called ‘tallies’ are structurally complex and are designed to represent completed enumerations; the Inka khipu knot records, for instance, record numbers using a decimal system with place-value (Ascher and Ascher 1980; Urton 1997, 2003). Many of the so-called tallies of medieval Europe are simply wooden slabs or blocks on which The cognitive and cultural foundations of numbers Roman numerals have been carved (Baxter 1989). ‘Tallies’ of this sort are simply numerical notations that happen to be notated on media different than those used for phonetic scripts. Tallies that notate quantities serially, using one mark for each object, however, are distinct from numerical notation. Thus, a tally for 15 might read IIIII IIIII IIIII. They are visual and representational, but the function of such artifacts is quite distinct: they are serial records of an ongoing enumeration activity rather than a final cardinal count (as in the Roman XV). Each sign, regardless of its shape or the spacing, represents one unit, and even though the signs can be read as a cardinal count, the process of making them is ordinal. They are immediate aids to computation, albeit of fairly limited flexibility. Because they are not intended primarily for permanent record keeping, their archaeological survival is limited, and in fact some tallies may not survive to be discarded. For instance, Herodotus relates an episode in which Darius of Persia tied sixty knots in a thong and then instructed a group of Ionian despots to untie a knot each day while awaiting his return (Herodotus, Histories 4.98). Tallying is evidently of great antiquity, and probably dates back at least to the Upper Paleolithic period (35,000–10,000 bc), when anatomically modern humans notched bones and possibly other perishable materials (Absolon 1957; Marshack 1972). In some cases, as in the Etruscan/Roman numerals, tally-systems gave rise to numerical notation systems (Keyser 1988). Nonetheless, by no means are tallies primitive, sub-optimal, or simply precursors of written numerals. Their functions are completely different; even in contemporary western societies all sorts of repetitive numerations are taken by marking tallies in groups of five, with the fift h crossing out the first four. Even in an age of widespread electronic computation there is no reason to believe that the humble tally, likely tens of thousands of years old, is at risk of disappearing. Nevertheless, tallying systems are primarily suited to serial counts of objects, rather than general arithmetical functions, and thus merchants and administrators generally require additional computational devices or representational systems to aid in arithmetic. There is substantial ethnographic and historical evidence demonstrating that computational techniques among non-literate or minimally literate groups are abundant and efficient for the tasks for which they are needed. While school arithmetic is generally decimal, the mental arithmetic of non-literate artisans and traders is often based on doubling, halving, and quartering (Petitto 1982; Rosin 1984). Basque-speaking shepherds in contemporary California use a highly effective array of computational techniques including spoken numeral words, mental arithmetic, and a tallying-system of pebbles and notched sticks (Araujo 1975). The Kédang people of Indonesia accomplish complex mensuration and numeration tasks, such as the measuring and evaluation of elephant tusks as part of the ivory trade (Barnes 1982). In various societies of Melanesia, ‘body-counting’ 503 504 people and practices is used in place of numeral words, naming various parts of the body in sequence as a means of counting (Saxe 1981; Biersack 1982). The pebble-abacus was the central technique for performing computations in the ancient eastern Mediterranean (Lang 1957; Taisbak 1965; Schärlig 2001). Roman and Greek numerals were often used to notate the column-values on the device, and to record the results of computations performed on permanent media, but otherwise were irrelevant to the practice of arithmetic. While only around thirty classical abaci have survived, Netz (2002, 327), noting that ‘The abacus is not an artefact; it is a state of mind’, rightly cautions that any flat surface and set of objects can suffice. Nevertheless, following the classical period, there is no substantial archaeological, textual, or artistic evidence for the use of the abacus between the fift h and tenth centuries ad. In the tenth century in Europe, a sort of abacus was revived under the particular influence of Gerbert of Aurillac (later Pope Sylvester II), who was also one of the primary early adopters of Arabic numeration in the West after being exposed to Arabic arithmetic during his travels in Toledo around 970 (Folkerts 2001). Instead of using multiple pebbles or balls in each column, Gerbert’s ‘abacus’ was a grid on which tokens called apices were laid out, each one bearing a western numeral from 1–9, or a zero-sign called tsiphra (Berggren 2002, 355–357). These tokens were manipulated by moving them from column to column, and its users were known as ‘abacists’. Yet Roman numerals predominated for actually writing the results of computations performed, and not until Leonardo of Pisa (Fibonacci) wrote his Liber abaci in 1202 did pen-and-paper arithmetic using western numerals begin to spread across western Europe, among the so-called ‘algorismists’ (Burnett 2006). Yet until the sixteenth century and the advent of printed arithmetics, most merchants and administrators used neither ‘Gerbert’s abacus’ nor Western numerals, but rather computation on boards with unmarked tokens or pebbles, much like the Greco-Roman abacus (Baxter 1989). Throughout the Middle Ages the English technique of choice was the cloth ‘Exchequer board’, etymologically related to ‘checkerboard’ (with results written in Roman numerals); the modern British title ‘Chancellor of the Exchequer’ preserves the linkage between the counting board and commerce (Murray 1978, 169). Another reliable, inexpensive, and portable computational ‘technology’ are the fingers. There is widespread evidence for the use of finger-numbering and arithmetic in classical Greece and Rome, including depictions of individuals reckoning with the fingers, tesserae ‘gaming tokens’ showing particular finger configurations along with Roman numerals, and abundant textual references (Alföldi-Rosenbaum 1971; Williams and Williams 1995). Finger-reckoning was the primary arithmetical technique employed in early medieval Europe, and was strongly praised by Bede in his work on calendrical computation (Wallis 1999). Finger-reckoning systems remained in use in Europe and the Middle East The cognitive and cultural foundations of numbers throughout the Middle Ages into the early modern period (Saidan 1996). Like any technique (including pen-and-paper arithmetic), finger-reckoning rests on a foundation of memorized arithmetic facts and/or visual representations such as multiplication tables. Chisanbop, an arithmetical technique developed in Korea in the 1940s, uses the fingers to notate and reckon as if they were a quasi-abacus (Lieberthal 1979). From at least the fourth century bc until the sixteenth century ad, East Asian arithmetical procedures were centered around the suan zi, or counting rods, and their written representation, the rod-numerals (see Volkov, Chapter 2.3 in this volume). The counting rods were thin sticks or strips of bamboo, wood, ivory, or bone, and could be manipulated in columns to represent numbers in a place-value, decimal manner, much as the Roman abacus. As with most technologies, the initial reaction to them involved some skepticism—the Daodejing, written around 300 bc, asserts that ‘Good mathematicians do not use counting-rods’ (Needham 1959, 70–71). Yet they were very quickly adopted, and were the foundation of Chinese mathematical practice until the late Ming dynasty. One of the primary advantages of the system was that the physical rods could easily be transformed into written numerals using horizontal and vertical lines to notate the position of the rods. Many Chinese mathematical terms use the radical for ‘bamboo’, further signifying the linkage between the counting rods and mathematics (Needham 1959, 72). Although physical rods themselves are no longer used, they survive in written form today in a numerical notation system called an ma, used in commercial contexts such as bills and invoices (Martzloff 1997, 189). The suan pan or Asian bead-abacus is of relatively recent origin, probably no earlier than the fourteenth century, and not until the seventeenth century did it definitively supplant counting rods. There is little evidence of a competitive environment between users of the suan zi and the suan pan to parallel the ‘abacist– algorismist’ debate in Europe or the later debate between users of the counting board versus users of western numerals. Nevertheless, the transition did occur throughout East Asia, where the suan pan (called soroban in Japanese) is a central part of mathematics education to the present day. No similar transition seems imminent today that would result in the abandonment of the suan pan. Although western numerals are ubiquitous in Japan and commonly used in China, Chinese numerals are rarely used for pen-and-paper arithmetic. This state of affairs is by no means indicative of a hidebound mindset or stubbornness. Stigler (1984) showed that Japanese master abacus users employ a ‘mental abacus’—a mental representation of intermediate and prior positions in a computation that greatly enhances the purely material aspects of the technology. A trained abacus user can normally manipulate multi-digit numbers far more rapidly than any reckoner using pen and paper. On 12 November 1946, the American military service newspaper, Stars and Stripes, sponsored a competition between Private 505 506 people and practices Thomas Wood, an American soldier trained in the use of one of the sophisticated electronic calculators available at the time, and Kiyoshi Matsuzaki, an administrator and abacus master (Kojima 1954). Although the competition was surely designed to impress the audience with the superiority of American technical ingenuity, the Japanese competitor won four of the five events. Zhang and Norman (1995) scorn abacus users for using one technique for arithmetic and another for writing results in numerals, as part of their argument that western numerals are uniquely efficient arithmetical tools. The merit of such a position vanishes once it is recognized that pen-and-paper arithmetic with western numerals cannot be demonstrated to possess this putative superiority. Although their surviving calendrical and divinatory texts do not notate calculations performed (only results), Landa reported in his Relación de las cosas de Yucatán that the Maya and related peoples of lowland Mesoamerica computed using a flat board or on the ground (Tozzer 1941, 98). The Guatemalan Maya at Panajachel in the 1930s reckoned using cacao beans or stones in groups of five or twenty, and this may be a survival of earlier Maya practices (Thompson 1941, 42). In sixteenth-century Peru, Don Felipe Guaman Poma de Ayala, the son of a conquistador and an Inka princess, depicted a khipukamayuq (khipu-administrator) using the traditional khipu system of knotted cords along with an abacus-like grid of black and white pebbles or stones (Wassén 1931; Urton 1998, 417–420). Because the khipu could not meaningfully have been manipulated for arithmetic, some abacus-like technique would have been needed to administer the expansive and multi-ethnic Inka Empire. Computational devices like the abacus were so prevalent in pre-modern states that one might reasonably ask why they would be replaced, given that, at the very least, they seem to have been as efficient as pen-and-paper computation. To answer this question, we need to look seriously at the alternative. The emergence and spread of numerical notation While many societies possess visual and/or material tallying techniques, only some societies possess numerical notation. Numerical notation systems are visual but primarily non-phonetic structured systems for representing numbers permanently. Typically they do so using a set between three and forty signs, which combine together by means of a numerical base, often but not always that of the language spoken by its inventors. Over 100 structurally distinct numerical notation systems are known to have been used between 3500 bc and the present day (Chrisomalis forthcoming; see also Cajori 1928; Smith and Ginsburg 1937; Menninger 1969; Guitel 1975; Ifrah 1998). Unlike number words, they represent numbers translinguistically, and do not follow the grammar or lexicon of any The cognitive and cultural foundations of numbers specific language. Unlike tallies, they represent completed enumerations, and unlike computational technologies, they create permanent records of numerals. They can be used for computation, but historically this function has been rare. The primary typological distinction among them is between additive and positional (place-value) systems, although this is not the only relevant distinction that can be made (Boyer 1944; Chrisomalis 2004). The earliest attested numerical notation is the proto-cuneiform system used in the ancient Mesopotamian city-state of Uruk in the late fourth millennium bc (Nissen, Damerow, and Englund 1993). In its initial state, proto-cuneiform writing consisted of a large repertory of at least fifteen different systems for numerical representations of different categories of objects, persons, and capacity measures, along with ideograms and pictograms for the various things being enumerated. It served as an administrative system for the urban temple economy of Uruk and other cities throughout Mesopotamia, and well as the Proto-Elamite area to the east, in modern Iran (Potts 1999). In a series of articles and books, the Near Eastern archaeologist Denise SchmandtBesserat has suggested that the Uruk numerical notations, and ultimately writing itself, are the end product of a millennia-long history of accounting and administration. Throughout the Neolithic in Mesopotamia, possibly as early as 8000 bc, clay tokens were used as administrative tools much as tally-sticks and knotted cords might, using one-to-one correspondence between the counters and the objects being enumerated, as part of a sophisticated accounting system. Schmandt-Besserat’s hypothesis, itself derived from the earlier work of Amiet (1966), is suggestive, but must be read in the context of severe criticisms such as those of Lieberman (1980) and Zimansky (1993). In particular there is little evidence that the specific forms of the clay tokens bear any resemblance to the proto-cuneiform signs. It would be erroneous, however, to assume that numerical notation developed independently only in Mesopotamia, or that the developmental trajectory that it took there provides a general template for its development elsewhere. In Egypt, the earliest numerals are found on labels for mortuary offerings, in a royal tomb from a cemetery at the city of Abydos, dating to around 3250 bc (Dreyer 1998). In China of the Shang Dynasty, the first written documents (c 1200 bc) are records of royal divinations (Tsien 2004). In the Middle and Late Formative periods in Mesoamerica (c 600 bc–150 ad), virtually all of the earliest Zapotec, Olmec, and Maya inscriptions contain numerals, but their use is strictly in names and dates, never administrative (Houston 2004). With the possible exception of the Egyptian case, there is very minimal likelihood that the development of numerical notation was spurred by diff usion from Mesopotamia or anywhere else. Rather, the development of both writing and numerical notation is correlated with the formation of early states in each region, but the functions for which these representational systems are used are quite distinct. 507 508 people and practices Postgate, Wang, and Wilkinson (1995) suggest that the reason we have not found evidence of early administrative writing and numerical notation in Egypt, Mesoamerica, and China is that such documents were written on perishable materials that have not survived. They use this line of reasoning to propose that writing emerges everywhere as it did in Mesopotamia: for bookkeeping and accounting-related functions concerning state administration. Yet in the absence of evidence that this is so, such assertions are unjustified. There is simply no reason to expect that the invention of writing and numerical notation must always have the same underlying function everywhere. It is nonetheless true that numerical notation developed independently only in socially complex societies that had considerable need to represent quantities. Nevertheless, in other cases—as in the expansive, densely populated states of West Africa—numerical notation simply never developed, so it cannot be regarded as an absolute necessity. Because numerical notation is used widely in exchange and administration, its spread and adoption is strongly correlated with imperialism, long-distance trade, and other political and economic processes associated with states. Although most numerals are used for representation rather than computation, the employment of numerical notation systems for astronomy, mathematics, and related scientific practices has also played a central role in their diff usion. Because numerical notation is not tied to any specific language, is not as difficult to learn as, for instance, a writing system, and is a communication technology used in the context of long-distance commercial and scientific exchanges, it diffuses readily in many circumstances. The well-attested spread of Hindu numeration to Europe through Arabic intermediaries has led diff usionist explanations to be widespread in the literature on numerals, often with good reason. I have argued, on the basis of cultural contact and structural similarities that the Greek alphabetic (or Ionian) numerals developed out of the Egyptian hieratic or demotic numerals used in the 6th century bce in the context of circum-Mediterranean trade relations (Chrisomalis 2003). Yet the Greek inventors of the alphabetic system were highly innovative; their use of the letters of the alphabet, in sequence, as numeralsigns was unparalleled elsewhere, and the uses to which Greek numerals were put differ substantially from those for which Egyptians used them. On the other hand the notion that most people are uncreative and therefore most mathematical developments made only once, and spread from a single center, is quite incorrect and frequently tinged with racist assumptions about non-European peoples. Seidenberg’s (1960; 1962) pronouncements on the diffusion of mathematics, geometry, and all numbers higher than two as part of a Mesopotamian or Indian Neolithic ritual complex, and his insistence that all Maya numeration and mathematics was derived from Babylonia (Seidenberg 1986), are extreme and unsupported by any textual or archaeological evidence. Joseph Needham’s remarks on the subject of the priority and diffusion of Chinese The cognitive and cultural foundations of numbers mathematics are more tentative and a necessary counterpoint to Eurocentrism, but nonetheless the notion that the Chinese spread place-value to Babylonia (Needham and Wang 1959, 146–150) or that Chinese mathematics influenced Mesoamerica (Needham and Lu 1984) cannot be sustained. In fact, it is highly probable that place-value numerical notation, or something quite like it, developed at least five times independently: in Middle Bronze Age Mesopotamia (c 2100 bc), in the Warring States period in China (fourth century bc), in lowland Mesoamerica (no later than 100 ad), in India (c 500 ad), and in the Andes (no later than 1300 ad). No two of these regions are less than 3000 km apart, and the development of place-value occurred centuries apart in each. Each development had antecedents in earlier, local notations and computational techniques, and each is distinct in various ways. For instance, the Chinese rodnumerals have a sub-base of 5, like the Roman abacus, and the Andean khipu notation lacks a sign for zero. The most parsimonious explanation for these developments is that place-value is more easily conceived than extreme diff usionists allow. This also provides support for the notion that mathematics is a pan-human activity whose foundations do not differ greatly among different societies. Numerals and computational efficiency A connection is frequently asserted between the present ubiquity of the Western numerals in worldwide usage and the utility of this system for performing basic arithmetic using pen-and-paper computation. At first glance, this hypothesis is extremely appealing. Many of the references to positional numerals by ‘early adopters’ explicitly praised positional numeration in comparison with other techniques and representations. The Syrian Christian bishop Severus Sebokht discussed Hindu mathematics in 662 ad, noting ‘their clever method of calculation, their computation which surpasses all words, I mean that which is made with nine signs’ (Nau 1910, 225–227). In introducing the system more broadly to the Middle East in the ninth century, the mathematician al-Khwārizmī promoted the use of the nine digits plus zero as an alternative to reckoning with the letter-numerals (hisāb al-abjad) with its twenty-seven alphabetic signs, or fingernumeration (hisāb al-‘uqūd), discussed above. Similarly, the western European debates between the abacists, proponents of the use of the medieval abacus with tokens, with numbers written in Roman numerals and algorismists, proponents of pen-and-paper arithmetic with western numerals, reflect contested narratives of efficiency. The algorismists struck hard with many positive evaluations of the western numerals’ efficiency, and eventually became predominant among mathematicians (Burnett 2006). Yet 509 510 people and practices conflict over computational techniques continued heatedly among users of the mercantile counting board as late as the seventeenth century. The famous allegorical representation in Gregor Reisch’s Margarita philosophica depicts Arithmetic, bedecked with western numerals on her gown, looking approvingly upon Boethius using western numerals while Pythagoras toils at his counting board with pebbles (Reisch 1503). Similarly, in his dictionary of 1530, the lexicographer-priest John Palsgrave included the sentence, ‘I shall reken it syxe tymes by aulgorisme or you can caste it ones by counters’ as a sample sentence for the verb ‘to reckon’ (Palsgrave 1530, 337).2 Nevertheless, we ought not to assume that the proclamations of advocates and early adopters perfectly reflected reality. The debate surrounding the adoption of place-value numeration, both in the Middle East and later in western Europe, pitted traditionalists against innovators and threatened to overwrite—literally— much of the practice of arithmetic, astronomy, mathematics, and accounting as they had been practiced for centuries. These debates were never solely about efficiency, but had significant ideological components. Struik (1968) argued that the prohibition of western numerals by the Guild of Moneychangers of Florence in 1299 was primarily part of the longstanding conflict between the Guelphs and Ghibellines in the mercantile economy of the city. The denigration of western numerals on the basis that they can be too easily altered, another often-heard reason for their prohibition, may have been a product of xenophobia against an Oriental invention. So, too, authors promoting the use of western numerals might do so, not solely on the asserted technical grounds, but because the promotion of a new arithmetical technique was part of broader social trends within late medieval society. The argument that the western numerals are computationally more efficient than Roman numerals for doing arithmetic is true, and continues to be raised by authors attempting to explain the decline of Roman numerals to their present vestigial use. The limitations of Roman numerals have been invoked as an explanation for the supposed impoverishment of Roman and early medieval accounting and mathematics (Glautier 1972; Murray 1978; Crosby 1997). The difficulty with this proposition is that Roman numerals, to our knowledge, were never used in written arithmetic in anything like the manner in which western numerals are, but of course through the abacus, through finger-computation, and through mental arithmetic. To a considerable extent, the preference for positional numerals is an artifact of modern western mathematics. Many additive systems of the past survived for millennia, such as the Egyptian hieroglyphic and hieratic numerals which persisted largely unchanged from the pre-Dynastic to the Roman period, suggesting 2. Ironically, though not unusually for the time, Palsgrave’s dictionary was foliated in Roman numerals. The cognitive and cultural foundations of numbers that they must have been perceived as desirable or useful for many purposes. There is a trend towards positional notation over time, but it is not inexorable and should not be presumed to now be irreversible (Chrisomalis 2004). In South Asia, the positional numerals ancestral to our own largely replaced the older additive Brahmi system between the sixth and eleventh centuries ad (Salomon 1998). However, in southern regions of the subcontinent, additive numerals continued to thrive alongside the Tamil, Malayalam, and Sinhalese scripts, right up to the colonial period (Guitel 1975, 614–617). The Tamil additive numerals continue to be used today for many purposes. Cultural resistance against the dominant traditions of northern India probably explains the retention of the additive numerals, but users of these notations suffered no evident disadvantage in their ability to undertake arithmetic. Similarly, the additive Chinese numerals would long ago have been abandoned in favour of the Tibetan numerals (a positional, decimal system transmitted from India) if this were the case. Of course, the reverse is true, for perfectly understandable reasons having to do with Chinese political domination in the region and throughout much of Central Asia. No one would consider efficiency for computation as the explanation in this circumstance. Yet the assumption that the western numerals predominate mainly due to their supreme utility, and that we have reached the timeless pinnacle of the history of numeration, remains commonplace in scholarly and popular works (Dehaene 1997, 101; Ifrah 1998, 592). A more parsimonious explanation for the current worldwide predominance of western numerals is the predominance of all sorts of western institutions, most notably scientific and economic, since the formation of the modern capitalist world-system with western Europe and later America firmly ensconced within the prestigious and powerful core (Wallerstein 1974). This process was accelerated by the early use of western numerals in printed books, in accounting documents, and on money—both the transmission of wealth and the transmission of information were governed by the western numerals. Changes in patterns of trade and intercultural communication correlate frequently with changes to numeral words as well as numerical notation; as the need for commerce and mathematics with larger, more complex societies increases, the numeral word series expands (Crump 1978, Schuhmacher 1975). The highest basic numeral word in European languages was ‘thousand’ until the thirteenth-century invention of ‘million’ among Italian bookkeepers, from which it spread to a wide variety of languages, Indo-European and otherwise. More significant was the fact that pen-and-paper arithmetic with western numerals produced permanent records of calculations performed, permitting errors to be perceived more easily. The regular practice of writing down calculations and their results also provided early modern landowners, merchants, and administrative officials a high degree of information and control over their 511 512 people and practices economic affairs (Swetz 1987). Nevertheless, much bookkeeping and arithmetic continued to be done without the aid of western numerals (Jenkinson 1926). As prominent a figure as William Cecil (Lord Burghley), Lord High Treasurer to Elizabeth I of England, regularly transcribed economic documents from western back into Roman numerals for his own convenience (Stone 1949, 31). Roman numerals were adequate if not optimal for recording results. The length of Roman numeral-phrases, often cited as a defect of the system, is only one of many factors users consider. Once the western numerals had achieved a critical mass of popularity among the newly emboldened European middle class, it became likely that others operating within the same economic and communication networks would adopt the system. New users adopted western numerals partly because the current users of the system were prestigious and wealthy—in terms of cultural transmission, this is a prestige bias (Richerson and Boyd 2005, 124–126). Their popularity allowed new users to transmit information to more individuals and thereby created a feedback system that further increased their popularity—a frequency dependent bias (Richerson and Boyd 2005, 120–123). Their utility cannot be conceived simply in terms of a structured system of signs, but also in terms of who and how many people were using them, and for what purposes. The property of frequency dependence is particularly notable in systems such as numerals for which communication is of central importance. It is linked to the ‘QWERTY principle’ explaining the persistence of sub-optimal but popular phenomena despite the existence of alternatives, and to the predominance of poor but popular recording media and computer operating systems. The ‘cost’ of not using the popular system is greater than the advantage of using the technically superior one. Conclusion At various times and places, individuals and groups may have adopted new numerical notations because of their perceived efficiency for computation. As a general explanation for the diff usion, adoption, and extinction of numerical notation systems, however, this theory is weak in comparison to the host of political and economic factors operant in any given social context. There is minimal evidence for the widespread use of written numerals as a computational technique prior to the development of Arabic numerals in the ninth century ad and their subsequent spread westward to Europe. Modern scholarly evaluations of the efficiency of various numerical systems for computation are interesting but irrelevant to their diff usion and extinction (Detlefsen et al 1975; Lambert et al 1980; Anderson 1958). When attempting to show these systems’ inferiority (as opposed to demonstrating the feasibility of such work), such analyses are The cognitive and cultural foundations of numbers perniciously derogatory. Moreover, with the growth of the electronic calculator industry over the past thirty years, pen-and-paper arithmetic may go the way of the slide rule before too many generations have passed. Because the relations between number words, arithmetical techniques, and numerical notation symbols are complex, the evaluation of the foundations of mathematics in any society is similarly complex. No two societies are alike, and yet the striking linguistic and cross-cultural parallels observed suggest that human thinking about numerals and arithmetic is highly constrained. The debate between universalistic and particularistic numerical systems will surely continue, as will the comparison of the utility of different numerical systems. 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