The Origins of Symbolic Number, by David M. O’Shaughnessy, Edward Gibson, Steven T. Piantadosi
MGH: Before we start this exhibit I’d like to paste an extract from an earlier exhibit of work by Ifrah which does not make a ‘cultural difference’ song & dance about the fact that some primitive groups today still do not have a word for ‘4’. One might argue (pace Nieder?) that these groups by definition do not present to us as high-performing fitness survivors in the relevant competitive Darwinian evolutionary processes and therefore cannot be expected to display highly evolved numeracy.
Contemporary “primitive” people also seem unable to grasp number considered in its abstract, conceptual aspect.
As a matter of fact, if a well-defined and fairly restricted group of persons or things interests the primitive ever so little, he will retain it with all its characteristics. In the representation he has of it the exact number of these persons or things is implied: it is, as it were, a quality in which this group differs from one which contained one more, or several more, and also from a group containing any lesser number. Consequently, at the moment this group is again presented to his sight, the primitive knows whether it is complete, or whether it is greater or less than before [Lucien Lévy-Bruhl]
"Primitive" people are thus affected only by a change in their visual perception, since they generally lack the abstract notion of the synthesis of distinct units…
… Early in this century there were still peoples in Africa, Oceania, and America who could not clearly perceive or precisely express numbers greater than 4. To them, numbers beyond that point were vague, general notions related to physical plurality. It is probably significant that, as Levy-Bruhl reports, some Oceanic tribes declined and conjugated in the singular, the dual, the trial, the quadrual, and finally the plural. Members of the Aranda tribe in Australia had only two basic number words: ninta ("one") and taia ("two"). For "three" and "four" they said taia-ma-ninta (“two andone”) and tara-ma-tara ("two-and-two"). Beyond tara-ma-tara they used a word meaning “many".
Islanders in Torres Strait, between New Guinea and Australia, had only these number words: netat ("one"), neis ("two"), neis-netat ("three," literally "two-one") and neis-neis ("four," literally “two two"); beyond that, they used a word meaning something like “a multitude”.
Among other examples of the same kind, we can mention the Indians of Tierra del Fuego, the Abipones in Paraguay, the Bushmen and Pygmies in Africa, and the Botocoudos in Brazil. When the Botocoudos said their word for "many" they pointed to their hair, as if to say, "Beyond four, things are as countless as the hairs on my head”.
The Source:
Here is today’s exhibit:
The Origins of Symbolic Number, by David M. O’Shaughnessy, Edward Gibson, Steven T. Piantadosi [2021]
Abstract
It is popular in psychology to hypothesize that representations of exact number are innately determined—in particular, that biology has endowed humans with a system for manipulating quantities which forms the primary representational substrate for our numerical and mathematical concepts. While this perspective has been important for advancing empirical work in animal and child cognition, here we examine six natural predictions of strong numerical nativism from a multidisciplinary perspective, and find each to be at odds with evidence from anthropology and developmental science. In particular, the history of number reveals characteristics that are inconsistent with biological determinism of numerical concepts, including a lack of number systems across some human groups and remarkable variability in the form of numerical systems that do emerge. Instead, this literature highlights the importance of economic and social factors in constructing fundamentally new [!?] cognitive systems to achieve culturally specific goals.
Introduction
One exciting hypothesis about human numerical cognition is that the origins of human mathematics can be found in innate systems of quantity representation. Indeed, there is abundant evidence documenting an evolutionarily ancient, cross-species ability to discern discrete quantities. These abilities have been demonstrated in a wide variety of animals ranging from insects to fish, birds, monkeys, and human infants even as young as 2 days old. Two behavioral patterns in perception have been documented: Precise discernment of small numerosities (known as subitizing; with a range roughly from 1 to 4), and an approximate and ratio-sensitive discrimination ability that obeys Weber’s law for larger numerosities, both of which can be derived from information-processing considerations.
In addition to these abilities (termed “quantical” cognition), many people also use symbolic resources for forming exact representations of large numerosities beyond the range of subitizing. This capacity can be seen in the familiar use of number words like “six hundred and forty thousand,” but, as we review below, language is not the only symbolic format found in human groups. The ability to symbolically represent exact large numerosities has been argued to transcend the other quantity representation systems because, on their own, systems for representing small numerosities are exact only up to about 3–5, and the psychophysics of the large quantity discrimination system is approximate. Neither, on their own, appears able to capture many people’s fluency with large symbolic numbers.
Here, we focus specifically on hypotheses about the origin of this system that can symbolically and exactly represent large numerosities. One foundational question in the study of numerical cognition has been to discover to what extent the conceptual knowledge for large, exact, symbolic number is innately available or biologically determined. Strongly nativist accounts provide a family of popular hypotheses to explain the origins of symbolic number.
We consider theories of numerical cognition to advance “biologically deterministic” or “nativist” arguments when they posit that: (a) innate quantity mechanisms scale up, essentially on their own, to produce large, exact symbolic number and perhaps mathematics (i.e., claiming that innate quantity systems have the requisite mathematical content, rather than merely functioning to provide input to learning mechanisms); or (b) there are innate mechanisms of representation that are isomorphic to the mathematical and logical foundations of number. …
… In this article, we draw on findings from both cognitive psychology and the anthropology of number in order to evaluate the position that human-like abstract number and mathematics primarily originate in biological mechanisms of quantity representation. The view that symbolic number arises through some strong form of biological determinism makes a number of predictions about the form and function of numerical systems across human groups.
Specifically, if the representations at the core of symbolic number acquisition are primarily biologically determined, we should expect that: (i) number should be found in all human groups; (ii) the mode of constructing quantity representations should be universally shared; (iii) number should emerge relatively easily in development; (iv) the timing of number acquisition should be roughly the same across human groups; (v) number should emerge easily within each human group; and (vi) the history of number should exhibit the level of abstraction hypothesized in nativist theories. It is worth thinking about each of these predictions in the context of behaviors which unambiguously are biologically determined, such as puberty or some aspects of the development of vision.
In contrast, we argue that symbolic number and counting fail all of the natural predictions (i)–(vi) of biological determinism. The history of mathematics shows that, large, exact, symbolic number is cognitively difficult for humans to create, the product of a long … history and that modern conceptions of number are far from universal across human groups [because not all contemporary groups are ‘modern’?]. The picture that emerges instead from the ethnographic literature reveals a diversity of … contingent approaches to problems of quantity, which are constructed according to local needs [which may be few], perceptions, practices, and history [which may be path-dependent]. …
Cross-Cultural Absence of Exact Number
Perhaps the most striking demonstrations of cultural influence on number are found in environments that do not afford the cultural support required to create any number words at all. The Pirahã culture provides a contemporary example, with number terms being extraneous to their mode of living and thus entirely absent from the language. Tasks requiring counting cannot be solved by native Pirahã people and the only words which express quantity can be shown to mean relative, not exact, quantity. For example, when Pirahã people were asked to count 10 physical objects in ascending order (i.e., from 1 to 10), they appeared to use a one-two-many count system [or maybe they were just having fun playing games with the researcher].
That is, participants labeled sets of size “one” as ho ́i, sets of size “two” as hoí, and all other sets as baágiso. However, when counting down, it was apparent that the terms were actually relative and approximate quantifiers rather than exact numerical terms—including even the word initially assumed to mean “one”. Specifically, the word that was thought to mean “one” really means “few,” and it can be appropriately applied even up to 5 or 6 objects if elicitation started with 10. Similarly, the word that was thought to mean “two” really means something like “some,” and the word that was thought to mean “many” does indeed mean “many.”
[Who is the one confused? the observer or the subject of observation?]
Thus the correct meanings only became apparent when the initial context was varied via the experimental manipulation of counting down. Beyond the Pirahã, these results raise the question of how many other cultures that appear to have small exact number words in actuality would be shown to have a similar relative system if the context were manipulated.
While Pirahã culture is remarkable in many other respects, other indigenous groups have used small number words in a nonexact manner. Hammarström (2010) surveys a number of languages that have been reported in the literature to have no exact number words above “one,” including two languages that potentially had no exact number words at all (Oro Win and Xilixana). John Peters, a missionary-turned-sociologist who spent many years living among the Xilixana during the initial contact period, described a numerically ambiguous system that “frustrated and infuriated” him:
Quantity is limited to three words, though the meaning can be modified by gesture. Mõle means one, and possibly two. Yaluku pèk means something between two and five, while yalami means anything more than two. : : : The highest amount would be indicated by using the term yalami together with a phrase meaning “like the trees of the forest.” They once told me that there were yalami people in a village they had just visited. I didn’t know whether the population was 16 or 80. This system obviously would not work in Western society for purchasing a bicycle or six items at the grocery store, but it was perfectly adequate for the Yanomami. Exact numbers were not important. (Peters, 1998)
Experimental work has shown that the Mundurukú of Brazil make use of quantity terms beyond “two” in an approximate fashion, and they likewise do not verbally count: When asked to name a quantity of dots on a screen (between 1 and 15 dots), Mundurukú people mostly showed consistency with terms for “one” and “two,” whereas terms for “three” and greater were not applied uniformly. For instance, when five dots were displayed, the term for “five” (pũg põgbi; or “one hand”) was used only 28% of the time. Other answers included “some,” “four,” “three,” “many,” or other idiosyncratic utterances.
This does not mean that quantity judgments are impossible for Mundurukú speakers, as exact small quantity and larger approximate quantity estimations seem to be universal and are easily demonstrated in the absence of learned number symbols. However, recent work by C. Everett (2019) has argued that even small numbers are not generally privileged in lexical systems of the world.
The use of numerical terms in a nonexact or vague manner has also been documented for a number of Australian indigenous groups, including one language where the term for “one” could be used approximately. The existence of vague number words in Australia suggests a lack of utility for precise numeration prior to colonization, although small numeral systems do not necessarily imply vague number usage. Yet as McGregor (2004) points out, in cases where collections with numerosities exceeding subitizing range are culturally designated as “many,” it makes little sense to apply an exact counting procedure since the numerically labeled quantities can be immediately and nonverbally apprehended.
Such cultural patterns would be surprising under theories in which all humans have an innate concept of “one” along with an innate successor function, since the concepts that these systems generate are not lexically marked in languages like these, but nearby quantity terms (e.g., “a few”) are. While the potential lack of utility for exact quantification historically might be invoked to defend strongly nativist accounts, such reasoning begs the question of why symbolic number (and especially large number) would be biologically encoded in the first place if it were not useful. Pirahã provides an interesting case in this context, where it is likely that symbolic number would be useful in trade, and yet it is neither adopted nor constructed [how much ‘trading’ do they actually do?].
In fact, Pirahãs rejected [actively, consciously?] number words and counting as outsider knowledge which is not required for a satisfying life. Explaining these cultural patterns remains challenging under strongly nativist accounts.
Forms of Number Representation Are Diverse Across Cultures
Many nonindustrialized people have developed rich numerical systems that are quite different from our familiar decimal counting system. Binary, quinary (base-5), and vigesimal (base-20) systems have commonly been encountered, along with comparatively rarer forms such ternary, quaternary, senary (base-6), and others.
But interestingly—and perhaps counter-intuitively—the concept of a single counting base is not sufficient for explaining the variety of systems that have been invented, with body-part systems and multiple bases also being common, thus necessitating new forms of classification. Body-part systems have been observed in South Eastern Australia, and the Torres Strait Islands, with Ray describing a base-2 verbal count coexisting with a 19-part body tally (which started on the little finger of the left side and looped over the body to the little finger on the right hand). [fitness test failed]
The names for the places were distinct from the numeral roots for “one” (urapun) and “two” (ukasar) however, as they were instead the literal names of positions on the body. Similar body-part systems have been documented in Papua New Guinea. Verbally these systems can be compared with modulus systems, although they may function like base counting systems when totals are able to be carried, either mentally or with additional bodies. An abundance of multibase systems have also been studied in Papua New Guinea and Oceania, for instance the common 2-5-20 cycle, where counting proceeds: 1, 2, 2 + 1, 2 + 2, 5 (often one “hand”), 5 + 1, 5 + 2, 5 + 2 + 1, 5 + 2 + 2, 5 + 5, and so on, up to a new base of 20 (often one “man”). Even systems of finger counting show a great deal of [bending?] diversity (Bender & Beller, 2012) …
While it may seem that the morphological labeling system for integers is a degree of freedom removed from the underlying semantics of a discrete infinity of integer concepts, in some cases morphological patterns seem inconsistent with notions of an innate recursive number generator.
[Did the researchers explain these concepts in these terms to the remarkable Pirahã respondents in the Amazonian jungle?]
For example, the theory of an innate successor function posits that each natural number is generated as the successor of a previous natural number, with recursion grounding out at “one”. This would suggest that addition by one—application of the successor function—should provide the most natural base system for counting since it would transparently map between words/morphology and meaning. However, even the way that count series are linguistically formed varies beyond the additive, as some systems exhibit multiplication and even subtraction in their construction.
Yoruba counting is one well-known system that employs sub- traction, with 15 root words and a primarily vigesimal structure. An example illustrating subtraction can be seen from the numeral words for 40–60, which in decimal form could be translated as: 20 × 2 [forty], 1 + (20 × 2) [forty-one], 2 + (20 × 2) [forty-two], and so on to forty-four, followed then by: −5 – 10 + (20 × 3) [forty-five], – 4 – 10 (20 × 3) [forty-six], et cetera up until: –10 + (20 × 3) [fifty]. To give a written example in Yoruba, “forty- five”would be márùúndínláàáḍo ́ta; where“twentyplacedthree ways” (the elisioṇo ́ta) occurs at the end of the phrase. Prior to this we have m (mode grouped), árùún (mode 5), followed by the elisions dín (it reduces), and then láàád (add 10 it diminishes; see Verran, 2000, p. 350). From fifty-one to sixty the system proceeds: 1 – 10 + (20 × 3) until fifty-four, then followed by: –5 + (20 × 3) and so on until sixty, or 20 × 3. According to Mann (1887), this particularly complex system may have a cultural genesis in the counting and distribution of cowrie shell currency, and in her earlier work Verran (2000) noted the etymological connections to the counter’s hands and feet.
[Very complex things these hands and feet — as Ifrah noted long ago the 10-base number system of the Incas was ‘invented’ by bending down to touch the toes.]
Variation in form goes beyond the structure of the counting system and includes its use. This may seem unfamiliar because, in cultures [societies?] with formal education, we may consider the ability to apply numbers to any set as one of their defining features. However, some cultures have imposed constraints on the category of items that their number words can be applied to, suggesting that the generalizability and abstraction that our culture finds in number—and encodes into features like an innate successor function—does not generalize well to other human groups. [Or they simply lacked the education/epistemology to know otherwise?]
Such diversity is particularly evident in Papua New Guinea, where variations likely developed through a combination of innovation and diffusion, moderated [or impeded?] by cultural constraints on necessity and interest in enumeration. For example, Ponam Islanders have an extensive decimal count system with terms upwards of 9,000, yet strikingly, not everything would be counted with this system. Carrier (1981) reported that as a rule, they did not count people:
Despite obvious skill with numbers, no one has any idea how many people live on the island, how many households there are or how many children are attending the primary school. Even more surprising, many parents of large families do not know how many children they have without stopping to think about it. And almost no one knows that there are 14 clans on the island, although everyone knows their names and can calculate the number in a few moments.
[It has been known for observers to be more gullible than the subjects of observation.]
Saxe (2014) has shown how the introduction of monetary exchange drives the creation of new forms of number representation and arithmetical abilities, but exchange itself does not necessarily imply number usage. For some groups the exact number of gifts given in a ceremonial exchange is important, whereas for others, it is the visual quality of the presentation that takes precedence, with exact number playing little to no role [Ah, the quaint old barter economy]. For the Ponam, it is not the absolute number of objects that matters in a ceremonial exchange, but rather the number relative to another’s gift (Carrier, 1981).
[MGH: The article continues at length to make the same culturalist arguments from various angles with huge numbers of rather uninteresting literature references which are time consuming for me to delete. So let us skip to the Conclusion …]
Concluding Discussion
Because we live in highly numerate and literate societies where virtually every adult understands not just counting but also at least some higher mathematics, it is easy to lose sight of the fact that throughout human history there has been a striking diversity of ways to represent number. Within the field of numerical cognition, the prevalence of theories based in biological determinism may be due to the fact that much of the work on number has prioritized animal work over data from nonindustrialized cultures. Animal work has been instrumental in understanding fundamental mechanisms of quantity representation across species, but nonhuman animals do not learn symbolic number in the same way as human children, nor do they have anything approaching the breadth of other mathematical abilities that are available to humans. Thus, in seeking the origins of number, it may well be that the differences between humans and animals ought to be emphasized, as opposed to the similarities.
[Reader: wherever you find the word ‘culture’ or equivalent undefinable concept try substituting words like ‘knowledge’, ‘cognition’ or the evolutionary biology terminology utilised by scientists such as Robin Dunbar and Andreas Nieder.]
A richly accumulative culture may be one of the primary differences between animals and all human groups. Depending on powerful and general learning mechanisms, cultural transmission permits mathematical knowledge to be passed down and perhaps leads to striking cascades in how people think. Formal and informal pedagogical practices prevent each generation from needing to reinvent mathematical abstractions (revising old ones instead), in what Tomasello (2009) called “the ratchet effect.” If part of this ratchet works to simplify and abstract, then over time cultural transmission may tend to remove complications that are otherwise perfectly natural for humans—such as counting systems that depend on the type of object, rely on subtraction, interface with physical objects, or use icons. If scientists happen to live in a culture where this ratchet has led to a particularly simple and abstract formulation of numbers (e.g., the Peano axioms), it would be easy confuse the outcome of this cultural process with human nature— especially if the breadth of human numerical creation was not well appreciated.
It is easy to forget, for instance, that the mathematics we know today—including concepts as simple as fractions, real numbers, and zero—has been unknown throughout almost all of human history, even up through the scientific revolution. [That is the point, some humans have remained isolated from such change, the scientific revolution has not reached them, the evolutionary pressure for numeracy has not impacted everywhere evenly, and culture has little to do with such unhappy facts.] Galileo formalized his physics with geometrical arguments, not algebraic equations, but generations of successive physicists have been able to reformulate the underlying ideas of physics in increasingly general and simple mathematical forms, from Newton’s laws, to Lagrangian and Hamiltonian mechanics, and more. Just as it would be a mistake to suppose that modern, elegant incarnations of physics must be innate because they are simple, it equally would be mistaken to think that distinctly WEIRD formulations of natural number somehow reflect a universal human nature.
Instead, the early ethnographic history suggests that representations are constructed ad hoc according to the [rather small and simple] cultural stock of available techniques—which are not necessarily attached to an underlying integer-like representation. Even when two domains are both plausibly numerical, some cultures have invented [or retained?] separate quantity systems which are not necessarily easily translatable. For the Kewa of Papua New Guinea, Franklin and Franklin (1962) write of a division between verbal counting and a body-part system used for specific purposes such as calendar reckoning: “The body-part system is not usually used to specify an exact number, e.g., one, five or ten. Informants cannot give the body-part system equivalent of a four base system number. Instead, the words meaning a few, lots, several, are used” [old news]. In terms of verbal counting, some indigenous groups have historically used count systems with relatively high practical upper limits, some have only linguistically marked numerosities of small sets within subitizing range, and others have not marked exact numerosities at all (at times excluding even an unambiguous numerical term for “one”). Iconic techniques, based on one-to-one correspondence, have allowed some groups to solve quantitative problems without the use of the symbolic number [or avoid having to solve very much at all]. The movement to mathematics in the current Western sense, and hence the study of numbers as abstract entities unto themselves, was dependent on the development of writing and the concerns of complex hierarchical state-like societies, and hardly a natural or universal occurrence.
Indeed, Chrisomalis (2009b, p. 427) argues that the most significant event in the history of numerical notation was not the often-cited invention of positional systems or the discovery of zero, but in fact the rise of capitalism as a dominant world system. [capitalism? how did that creep in to this discussion?]
Importantly, our claim here is not that evolution and biology play no role in symbolic number learning—clearly humans differ from other animals in their capacity for mathematical and numerical cognition, and this difference could not exist without a biological foundation. As in any other cognitive domain, comparison between humans and other animals is likely to be informative, for instance, about how evolutionarily ancient systems shape our own learning and perception. Our claim is that whatever biological heritage is relevant for large exact numbers, the biology does not determine the form of mental numerical content. We have argued that this is evidenced in the diversity of numerical systems that exist, the cultures in which they do not, and the difficulties of constructing number both historically and developmentally.
The history of number shows that appeals to nativism—specifically innate content tantamount to the integers—oversimplify and obscure crucial sociocultural processes [or the lack of knowledge/education?]. Thus, although our focus here has been on cultural factors, it is apparent that the correct picture is in fact a biocultural one [please define!]. It is likely that biological learning mechanisms can be broadly deployed to learn structures and procedures across domains, supporting the variety of numerical forms that have emerged cross-culturally and throughout history (as well as their absence).
This points to the need for developmentalists to pursue theories of how procedural knowledge may be learned, represented, and taught; perhaps drawing on our ability for metaphor and structure more generally. One approach to this is to formalize statistical learning theories that work over general spaces of algorithms like those in counting and other domains because children learn counting as an algorithm. Indeed, developmental theories of early number should have drawn on the now classic educational literature showing that children have a rich ability to revise and improve algorithms that they have been taught in domains like addition. Models of such learning argue that children effectively choose between known procedures or create fundamentally new procedures. If such abilities are available early in learning, and people’s capacity to infer such procedures extends far beyond numerical content, it is then most natural to hypothesize that number learning itself depends on these much more general algorithmic mechanisms of learning and representation. [Good idea, fund and set up schools in Amazon jungle, give the graduates formal qualifications, offer them jobs, watch them depopulate and turn their back on precious ‘culture’ in favour of high number $ quantities, Mexican border, and an abundance of capitalism].
Nativist developmental theories have tended to argue that specific innate content is required for large exact representations, while ignoring the fact that developmental processes can acquire so much more—including fractions, real numbers, complex numbers, vectors, matrices, tensors, arbitrary groups or fields, transfinite numbers, logic, set theory, calculus, non-Euclidean geometry, computability theory, and so on. Considering the breadth of human learning ability, theories which posit that exact symbolic number is determined by innate resources—while accepting these other do- mains as constructed—would seem to miss the marks of parsimony and adequacy, in addition to failing the most natural empirical predictions we describe above.
[You have now reached the end of this Social Science Files exhibit.]
[Encounters with culturalism are always tedious but necessary to remind ourselves of why we seek out and stay with the serious writers (e.g. among the most recent … the above mentioned Robin Dunbar and Andreas Nieder [Archive]). I do apologise for my impatience with it. Here among the Aboriginals in Australia it is now bedtime!]
The Source of today’s exhibit has been:
O'Shaughnessy, D. M., Gibson, E., & Piantadosi, S. T. (2022). The cultural origins of symbolic number. Psychological Review, 129(6), 1442–1456. https://doi.org/10.1037/rev0000289
Social Science Files displays multidisciplinary writings on a great variety of topics relating to evolutions of social order from the earliest humans to the present day and future machine age.
‘The Heller Files’, quality tools for Social Science.