Sounding the depths at the confluence of numerosity and language

1 Introduction

One of the distinguishing characteristics of our species is the ability to think with numbers in precise ways. Most adult humans can, for instance, precisely differentiate large quantities in ways that members of other species apparently cannot. This ability has naturally transformed the human experience by enabling mathematical thought of all kinds, and ipso facto, facilitating socio-economic, technological, and other modes of behavior that shape cultural ecologies and the day-to-day lives of individuals. Yet, as is becoming increasingly apparent in current studies on the nexus of human numerosity and language, this ability and other interrelated ones are, contra some popular assumptions, not endemic to the human condition. Instead, such studies are revealing that these abilities are themselves a byproduct of linguistic and cultural innovations that, while overwhelmingly recurrent among people groups, are not universal. As we will see below, for instance, current work suggests unequivocally that number words are “conceptual tools” (Wolff and Holmes 2010, Frank et al. 2008; Wiese 2003) that augment human thought in pivotal ways.

2 Some historical background on numbers in Amazonia

Anecdotal evidence for the claim that the mere differentiation of some quantities is not evident in the behavior of all cultures comes from inspections of the historical record of Amazonia. Consider the following insight into the history of the Tapajós river, one of the main fluvial highways of the world’s largest rainforest:

The Mundurukú was a large and formerly warlike tribe on the upper Tapajós river that took to tapping rubber – and they still do to this day. Working hard, the Mundurukú gathered large quantities of rubber. But they were woefully uncommercial, easily duped because they did not understand arithmetic. Regatão traders sold them goods at a fourfold mark-up, including quantities of cachaca rum and useless patent medicines. They were of course paid poorly for their rubber.

(Hemming 2008: 181)

The Mundurukús’ history of struggling with mathematical concepts (even when contrasted to other Amazonian tribal groups without mathematical education) is in some sense then a matter of historical record – though the motivation for their struggle, viz. their lack of precise numeric language, has only been elaborated in recent linguistic and psychological fieldwork. In a similar vein, speakers of Pirahã, related to the language (now extinct) of another major once-warlike tribe, the Mura, have for some time been known to struggle with mathematical concepts. Unfortunately but predictably, this struggle has also been exploited by traders along the Marmelos and Maici rivers in central Amazonia, and has been witnessed first-hand by a number of missionaries working in the Pirahãs’ midst since the mid-twentieth century. From the time of their first contact with non-indigenous persons in the late eighteenth century, the Pirahã have evinced a well-documented and understandable xenophobia, having rejected nearly all foreign items both material and behavioral (D. L. Everett 2005). This rejection includes the failure to culturally incorporate technologies such as writing and, perhaps more surprisingly, number words. In contrast, such technologies have been readily adopted by many of their neighboring autochthonous groups.

There is no evidence that the Pirahã or Mundurukú are somehow genetically aberrant, and that their documented difficulties with basic mathematics stem in any way from innate cognitive obstacles. There is also no evidence that their comparative under-reliance on number terms is due to some ecological determinism since other Amazonian groups, many with similar subsistence patterns, have precise number terms. Arawak, Carib, Tupi, and Ge languages, along with others from smaller stocks, have number systems– some quite elaborate. This is particularly true in the case of Arawak but is also true in languages of other families. For instance, the Tupi-Karitiana, with whom I have conducted much of my own fieldwork, have an elaborate quinary/vigesimal system that can be used for representing quantities well in excess of 100. Some native Amazonian number systems are just being uncovered, for instance the binary/quinary based number system of the Jarawara (C. Everett 2012). In addition, there is now archaeological evidence that number systems and associated precise numerosity date back many millennia in Amazonia, as evidenced for instance by one cave painting, with regular markings of ‘eg-like symbols, amongst the many uncovered near Monte Alegre, Para. (Some of the paintings near Monte Alegre date back over 12 millenia.) The central portion of the painting in question is depicted abstractly in Figure 1. As is evident in the figure, there are clear correspondences between rows and columns demonstrating the appreciation of one-to-one matching for quantities at least as high as six. Recent results discussed below suggest such an appreciation would be implausible without numeric language. Of course it is not particularly surprising that numeric language and concomitant precise numerosity is ancient in Amazonia as it is elsewhere.

Figure 1

Abstract depiction of central portion of cave painting in Monte Alegre, Para

In addition, a series of geometrically regular geoglyphs dating back well into the pre-colonial era, some as far back as 600 BCE, have been uncovered in southwestern Amazonia (Schaan et al. 2007). For instance, there are square geoglyphs with very regular sides, some approximately 200 m in length, and angles. The function of these geoglyphs remains a matter of investigation. Four of the many geoglyphs are clearly visible in the one-minute satellite flyover “tour” in S1 (supplemental material). Two pictures of geometrically regular glyphs are provided in S2 and S3. It is highly implausible that these and other geoglyphs in the region could have been constructed without access to precise numerosity and means of recording quantities. Given that evidence for precise numerosity has been carved deep into Amazonia’s history, it remains a mystery why and how some groups of the region utilize typologically remarkably modest systems of precise number, or how a group like the Pirahã can lack one altogether.

The people in question are “intelligent”, as evident by their successful adaptations in local ecologies. Like the adaptations of many other Amazonian groups, these have long baffled foreigners struggling to survive in those same environs. Because of their status as well-adapted adults who also struggle with mathematical concepts and, crucially, lack any exact number terms (in the case of the Pirahã) or many such terms (in the case of the Mundurukú), the two aforementioned groups have drawn an inordinate amount of attention from cognitive scientists in the past decade. The resultant research conducted among these and a handful of other groups has been aimed at clarifying the relationship between human numerosity and numeric language. The goal of the work is not to lionize or mystify the superficially atavistic nature of such cultures though, perhaps unfortunately, the results of such studies do occasionally lead to such representations in the media. The aim of such work is also not simply to shed light on esoteric cultural patterns or exoticize remote tribal groups. Instead, the purpose of such studies is to look through a well-placed window toward the interaction of language and cognition among all humans, by considering carefully the extant test cases in which people choose to live without, or with few, numbers. Such cases are pivotal to our understanding of the role of language in contemporary human numerosity (C. Everett 2013a), and pivotal to shedding light on the role that cognitive technologies (Frank et al. 2008) like number terms have played in our species’ development. In the remainder of this paper, we will encapsulate some of the recent findings of such work. We will suggest that the work in question is illustrative of the way linguistic fieldworkers are uniquely placed to make contributions to pivotal debates in the cognitive sciences.

3 Current research

There is extensive ontogenetic, phylogenetic, and cross-cultural evidence that human infants (Wynn 1992; Xu and Spelke 2000) and other species (Feigenson et al. 2004) are genetically endowed with two primal strategies for thinking about numbers. One of these is the approximate number system or ANS (see e.g. Dehaene et al. 1999), used to estimate quantities in a fuzzy manner. This system is evident in estimations of quantities characterized by Weber’s Law, according to which the size of estimation errors increases in proportion to the size of the actual quantity being estimated. Such errors are characterized by a consistent coefficient of variation (standard deviation of estimation errors divided by the mean number estimated) for each quantity being estimated.

In addition, humans and many other species are innately equipped with the capacity to exactly differentiate quantities less than four, a capacity we rely on for subitization. Unlike other species, however, most adult humans are capable of precisely appreciating and differentiating quantities greater than three, apparently because we are able to combine our two innate quantity recognition strategies served by distinct neurophysiological substrates. (See survey of evidence in e.g. Feigenson et al. 2004 and Lemer et al. 2003.) Until the last decade, it was unclear how dependent this unifying capacity was on numeric language such as cardinal numbers and counting strategies. Since humans in most cultures mature and are inculcated with numerical language simultaneously, distinguishing general maturation and acculturation effects from linguistic ones seemed a near impossibility.

Languages can denote quantities via lexical items or morphosyntactic strategies. In some cases they may utilize both the former and the latter, in other cases only the former, and in others they rely principally on the latter (for instance many aboriginal languages of Australia have ubiquitous markers of grammatical number but a paucity of numerical lexical items). But in the case of the Mundurukú and in particular the Pirahã, standardized elicitation methods (Pica et al. 2004; Frank et al. 2008) have demonstrated that numeric language is fuzzy. Pirahã in particular lacks all forms of grammatical number and evinces no precise number terms. There are three Pirahã terms associated with approximate quantities, namely hói (used for one but also potentially other small quantities), hoí (something like “a couple” or “a few”), and baágiiso (many). Audio files of native speakers producing clauses with these latter words are provided in S4 and S5. In short, Pirahã is the most anumeric language documented, though other uncorroborated claims of anumericity are evident in the literature, for instance the Amazonian language Xilixana Yanomamo. Furthermore, many other languages have very modest number systems. (See survey in Hammarström 2010)

The most recent work amongst the Pirahã suggests that they struggle with even basic quantity recognition and differentiation skills, even the mere recognition of exact correspondences for quantities greater than 3 (C. Everett and Madora 2012). In keeping with Gordon (2004), the results of such work suggest that speakers of the language employ the ANS when attempting to match quantities presented to them in arrays, and this is apparently true regardless of the modality in which they are presented stimuli (C. Everett 2013b). Furthermore, contrasts of the results in C. Everett and Madora (2012) and Frank et al. (2008) suggest that Pirahã exposed to innovated terms for exact quantities exhibit improved performance on quantity recognition tasks. Such results suggest quite strongly that the mere differentiation of exact quantities greater than three (and an understanding of the successor principle), in addition to the manipulation and storage of such quantities (Frank et al. 2008), relies on the conceptual tool of numeric language. As noted in C. Everett (2013a, 2013c), these results are broadly consistent with results obtained among Nicaraguan home signers that do not have exact terms for cardinal sets either (Spaepen et al. 2011; Flaherty and Senghas 2011), and are also consistent with subtraction-based tasks carried out amongst the Mundurukú (Pica et al. 2004).

Figure 2

The basic one-to-one matching task. Participants are asked to match stimuli with a target array, after the task is modeled. Performance is consistent with usage of the ANS, rather than precise differentiation, for quantities beyond three.

Figure 2 depicts a trial of the basic one-to-one recognition task, in which Pirahã were tasked with recognizing mere correspondences between quantities by matching a stimuli array with their own array. Their performance on this task and related ones, some with different stimuli and in different modalities, deteriorates progressively for quantities greater than three. The related tasks include an orthogonal matching task, in which the stimuli array is presented orthogonally vis-a-vis the edge of the table in front of the participant, and a hidden matching task, in which the stimuli array is covered by a piece of cardboard after being presented for several seconds. The deterioration of performance in these tasks is evident in the results presented in Figure 3. Those results, like those in Gordon (2004) and most of those in Frank et al. (2008), suggest that, absent the conceptual tool of precise numeric language, speakers of an anumeric language are unable to extend the precise differentiation of lower quantities to larger quantities. That is, they appear to be unable to link the two innate neurophysiological capacities for quantity recognition. Some exposure to innovated number terms seems to have benefited the Pirahã in one village, however (See C. Everett and Madora 2012).

Figure 3

X-axis values represent the number of stimuli in target arrays presented to participants.

Scale on left of y-axis represents proportion of correct responses. Scale on right of y-axis represents coefficient of variation (standard error of responses divided by response mean, for each quantity presented). Proportion of correct responses deteriorates for quantities beyond three, for all three tasks. Coefficient of variation hovers around 0.15 for all quantities, however, suggestive of analogue estimation – even in the case of the one-to-one matching task. Adapted from C. Everett and Madora (2012).

The above encapsulation necessarily overlooks some points of contention in the relevant literature regarding the manner in which number terms impact basic thought about quantities, as well as the extent to which particular linguistic practices such as counting play a role in enabling such numerical thought. Furthermore, it should be acknowledged that nonlinguistic cultural practices and symbols can also facilitate quantity processing and it is unclear how much of a role such nonlinguistic strategies could play absent numeric language.^[1] Consensus has been reached about the following major point, however, due in part to the work among the Pirahã and Mundurukú: “how a language represents large exact quantities dramatically influences how its speakers are able to store and manipulate them” (Frank 2012: 234). This claim is consistent with the data from anumeric and nearly anumeric populations, as well as ontogenetic findings from numeric societies (e.g. Condry and Spelke 2008; Carey 2001). This pivotal interaction between language and numerosity can be taken as a clear instance of linguistic relativity, then, since crosslinguistic variation in this semantic domain clearly yields cross-cultural disparities in cognitive strategies vis-a-vis quantity storage and manipulation. Number terms seem to play a fundamental role in shaping numeric ontologies (C. Everett 2013d).

In a similar vein to such findings on quantity recognition, Dehaene et al. (2008) report a series of experimental results demonstrating that Mundurukú speakers estimate numbers logarithmically, when tasked with spatially mapping perceived quantities of stimuli (both visual and auditory), along a line. Such work suggests as well that number words, which the Mundurukú do not utilize with high degrees of precision, are crucial to the development of non-logarithmic methods for the linear representation of quantity set sizes.

Other results collected among the Mundurukú have made substantive contributions towards our understanding of the interaction of numerosity and language, beyond supporting the notion that number words assist in the exact differentiation of larger quantities. Mundurukú speakers have imprecise words for numbers but, unlike the Pirahã, do have words that denote cardinal sets of one, two, three, and four, respectively, at least in the majority of cases in which such terms are elicited. In contrast, the Pirahã quantity related words do not denote such specific quantities even in a majority of elicited cases. (This cross-group disparity is evident when contrasting the results of the elicitation tasks in Frank et al. (2008) and Pica et al. (2004).) The Mundurukú have numbers for sets with one (pug/pug ma) and two (xep xep) items, and these are elicited in nearly 100% of cases in which subjects are asked to describe such sets (Pica et al. 2004). McCrink et al. (2012) demonstrate that, despite the generally anumeric nature of Mundurukú, its speakers are as capable at halving large quantities via approximation as are control French speakers. Such work suggests that the ANS can be implemented to halve quantities regardless of a native speaker’s language. Nevertheless, it is worth noting that the results have not been corroborated with speakers of a language without any precise number terms.^[2] After all, given that Mundurukú has words for one and two, unlike Pirahã, we might expect them to better utilize the concept of halving.

Another noteworthy finding recently made amongst the Mundurukú is that the accuracy of the ANS is heightened by increased access to mathematical education. That is, the abilities of speakers to estimate quantities with acuity is contingent in some part on exposure to math, not just on normal maturation processes (Piazza et al. 2013). Such work is consistent with the notion that cultural practices outside the linguistic realm impact the usage of the ANS. Nevertheless, it is an open question how much of the documented benefit of mathematical education to the ANS is due to continued exposure to numeric language associated with such education.

4 Future paths

Recent work on Polynesian languages has demonstrated that the number bases used by small cultures can impact their members’ numerical cognition in other unexpected ways. Speakers of Tongan and Mangarevan, for instance, traditionally had access to counting systems with different bases, and the choice of system varied in accordance with the type of object being counted (Bender and Beller 2013; Beller and Bender 2008). Yet there is a unifying abstract logic to the different kinds of counting systems within languages like Mangarevan and Tongan, which suggests the variant systems evolved to facilitate numerical processing in different cultural contexts (Beller and Bender 2008). Similarly, other work has demonstrated that variations in the kinds of numerical systems can impact quantity grouping and differentiation strategies, and suggests that some putatively primitive binary-based systems actually present significant cognitive advantages (Bender and Beller 2014).

Such issues remain largely unexplored in Amazonia, and one hopes that fieldworkers will be able to address them in the coming years, particularly given the moribund states of many Amazonian languages. In addition, many comparatively vibrant Amazonian languages have endangered numerical systems since Portuguese and Spanish number words predominate due to their greater transactional utility. Such homogenizing factors aside, a perusal of grammars of Amazonian languages reveals a host of variant bases and number types whose cognitive import has been largely unexplored. Yet descriptions of Amazonian languages often pay little attention to native number systems, beyond occasional number word lists, in part since such systems have often undergone partial or substantive replacement by national languages. In some cases this replacement can lead to erroneous assumptions regarding native number systems. For instance, it was claimed that the Jarawara language had no native number systems (Dixon 2004), a claim that proved inaccurate upon closer inspection (C. Everett 2012). In fact, that language has a robust binary-quinary system. This binary-quinary system only surfaced after systematic elicitation, and such documentation could reveal other faulty assumptions about numeric language in Amazonian groups. Systematic number-term elicitation will hopefully become a more standard implement in the linguistic fieldworker’s toolkit and contribute to our growing awareness of the typology of numbers. (See e.g. Evans 2009 and Donohue 2008 for some recent findings.) Such elicitation should ideally be carried out in multiple villages of the given people groups, since the assumption of homogeneity of numeric linguistic practice has turned out to be erroneous in two notable cases in Amazonia (C. Everett 2012; C. Everett and Madora 2012).

One additional benefit of such research, beyond linguistic description and beyond the elucidation of interactions between language and cognition, is the contribution towards our understanding of the diachronic development of number systems. For instance, Epps’s (2006) fascinating delineation of native number terms in Nadahup languages demonstrated the social basis of number-term development in that small family, and C. Everett (2012) presented data demonstrating the existence of native number terms in proto-Arawa. Like the aforementioned archaeological studies, then, such work can help to better establish the history of numbers in Amazonia.

Finally, work at the nexus of numeric language and cognition can also contribute to our understanding of the interaction of language and culture. Countless material technologies are dependent on the ability to precisely differentiate quantities greater than three, and many cultural practices are obviously dependent on the innovation or adoption of number words and counting. This finding is consistent with the worldwide correlation, albeit a weak one, observed between a culture’s subsistence strategy and the complexity of its number system (Epps et al. 2012). We would not predict a pervasive association between these factors, in part since almost all groups have some number system that allows, at the least, for the differentiation of small quantities. Nevertheless, it is worth stressing that there is in fact some correlation between “hunter-gatherer” subsistence and limited numeric language.^[3] Further explorations are required to better understand potential correlations between linguistic and cultural patterns associated with number systems. Given the results discussed here, we might expect that certain material technologies associated with agriculture are unlikely to develop in the absence of numeric language.

5 Conclusion

There is still much work to be conducted, in Amazonia and elsewhere, to shed light on the ways in which numeric language influences thought about quantities. The experimental methods used in such work are not generally elaborate and often require only simple tasks and associated statistical tests. In contrast to the work required of typical linguistic fieldwork, e.g. immersion in a language, extensive transcription, months of living with a people group, etc…, the research burden associated with such experimental work is comparatively modest. Most studies on this topic have been conducted by psychologists who are very much reliant on the knowledge of linguistic fieldworkers. Linguists with training in experimental methods are uniquely placed, then, to make their own contributions to such discussions, working collaboratively when possible and independently when necessary. In addition to carefully describing the number systems of dying tongues via careful elicitation and concomitantly shedding light on the development of numeric language in particular families (assuredly useful aims in their own right), linguistic fieldworkers in Amazonia will hopefully make vital contributions to our understanding of how language influences and enables quantitative thought. There is some urgency here, given the endangered state of many of the languages in the region, and given the increasing presence of socio-political obstacles to such work in Amazonia.^[4]

In short, fieldwork in Amazonia can contribute and has contributed important brushstrokes to the increasingly clear depiction we have of the interactions between human language, human thought, human culture, and human history. Here we have focused on one part of this tableau, the nexus of language and certain kinds of thought. Yet there are many other parts of the confluence of numerosity and language that are yet to be investigated. Linguists and others will hopefully continue to sound the depths of this confluence in the coming years.