As Easy as One, Two, Three?—Or Can the Pirahã Count?
“an almost complete lack of numeracy, an extremely rare linguistic trait of which there are only a few documented cases. The language contains no words at all for discrete numbers and only three that approximate some notion of quantity—hói, a ‘small size or amount’, hoí, a ‘somewhat larger size or amount’, and baágiso, which can mean either to ‘cause to come together’ or ‘a bunch’. With no way to express exact integers, the obvious question is: How do the Pirahã count?”
The short answer is that they don’t. According to a recently published paper in Cognitive Science by the anthropological linguist Caleb Everett, the Pirahã never engage in counting. Caleb Everett’s father, Dan Everett, the foremost expert on the Pirahã language, agrees. But in actuality, the situation is not that simple.
In his earlier work, Dan Everett listed several quantity words in Pirahã, including the two cited by Vuolo, with approximate quantity meanings; Everett, however, originally translated these terms as numerals: hói ‘one’ and hoí ‘two’ (the two words differ only in the tone, indicated by the accent mark). The word baágiso was originally translated by Everett as ‘many’—it can occur only with count nouns, such as ‘canoe’ and ‘arrow’. Its counterpart with mass nouns, such as ‘water’ and ‘sand’, is xapagí/[xogií] ‘much’. Additional quantity words listed by Everett include xogió ‘all’, xoíhi ‘a little’, xaibóai ‘half’, and xi ába ‘nothing’. In his later work, however, Dan Everett changed the translation of the two crucial quantity words, rendering them as ‘small size or amount’ and ‘somewhat larger size or amount’. But any conclusions about the nature of the Pirahã vocabulary and grammar depend crucially on how hói and hoí are translated, an issue that cannot be easily resolved given that most other scholars have only indirect access to the language and are limited to reanalyzing the data collected in the field by the Everetts and a few other scholars.
Everett’s claims about Pirahã numeracy go beyond issues of lexicon and grammar, as he has argued that the lack of words to express exact integers is a product of certain gaps in Pirahã culture. Clearly, the Pirahã live in a very different world, one that is not nearly as data-driven and number-laden as ours. Their “hunter-gatherer society with limited commerce is precisely the kind of environment in which a ‘small number system’ might be expected to survive without pressure to grow larger” (Nevins, Pesetsky & Rodriguez 2009: 386). But Everett’s specific claim is that in Pirahã “communication is restricted to the immediate experience of the interlocutors”. But it is not clear how the paucity of number words indicates a limitation to “immediate experience”: why would ‘large canoes’ be part of the immediate experience for the Pirahã but not ‘three canoes’? Also, if Everett’s later translation of the quantity words mentioned above is correct, ‘somewhat larger amount of canoes’ must be part of the immediate experience, unlike ‘three canoes’.
In order to address this controversial issue, several other researchers have done experimental work with the Pirahã. In a so-called “ten spools experiment”, Pirahã speakers were presented thread spools added to the array one at a time (the ascending condition). When shown just one spool, the Pirahã speakers always answered hói, which is consistent with the numerical interpretation of this word as meaning ‘one’ rather than ‘small size or amount’. This hypothesis is further confirmed by the fact that these speakers also responded with hoí ‘two’ when shown two spools. When more spools were added, they answered with either hoí ‘two’ or báagiso ‘many’. This use of hoí to describe more than strictly two objects suggests that the interpretation in Everett’s later work is correct. Thus, the ascending condition provides data that is partially compatible with one hypothesis and partially with the other. The evidence gets even messier when spools are taken away one by one (the descending condition). When shown from 10 to 6 spools, the speakers responded with either hoí ‘two’/‘somewhat larger size or amount’ or báagiso ‘many’. One speaker even responded with hói ‘one’/‘small size or amount’. Five or four spools were described as hoí ‘two’/‘somewhat larger size or amount’, and three to one spools as hói ‘one’/‘small size or amount’. Most importantly, the same number of spools did not always elicit the same response in the ascending and descending conditions, suggesting that the Pirahã words denote approximate quantities rather than precise numbers. So far, no sensible explanation has been proposed for this discrepancy.
Several other experiments, designed to shed some light on the use of number words in Pirahã, add food for thought and fuel to the debate. One such study, published by Peter Gordon, a psycholinguist at Colombia University, in 2004, addressed the following question: Can speakers of Pirahã appreciate larger numerosities without the benefit of words to encode them? According to Gordon,
“numerical cognition is clearly affected by the lack of a counting system in the language. Performance with quantities greater than three [is] remarkably poor, but [shows] a constant coefficient of variation, which is suggestive of an analog estimation process.”
As the first part of the study, Gordon replicated the “ten spools” experiment, where the Pirahã had to name the number of objects and use their fingers to supplement oral enumeration. The Pirahã speakers did poorly even for ‘four’. The results are schematized in the table on the left: not only was the use of words like hói ‘one’ and hoí ‘two’ inconsistent with their interpretation as “number words”, but the confused showing of fingers seems to indicate that the Pirahã have trouble appreciating numbers larger than three. Gordon also noted that “there [is] no recursive use of the count system—the Pirahã never used the count words in combinations to designate larger quantities.”
In addition, Gordon conducted a series of experiments in which the number of objects (thread spools, AAA batteries, uninflated balloons, ground nuts) presented by the experimenter had to be matched by the same number of objects laid out by a Pirahã speaker. The picture on the left illustrates one of the conditions, the so-called Orthogonal Matching: the experimenter’s array was lined vertically (away from the Pirahã speaker), and the subject’s “replication” was lined horizontally, right to left. This is task C in the chart of the results below.
In this diagram, the experimenter’s side is shown below the line and the replication above the line. Tasks A through D required the participant to match the array presented by the experimenter using a line of batteries or other objects. Task A was the simplest, as the experimenter’s array was an evenly spaced line of objects, which the Pirahã speaker has to match one-to-one. In Task B, the objects on the experimenter’s side were placed in a cluster rather than a straight line. Task D involved an unevenly spaced line-up of the objects on the experimenter’s side. Task E entailed a task, unfamiliar to Pirahã speakers, of copying lines drawn on paper. Task F was a matching exercise in which the participant saw a numerical display for only about a second before it was hidden behind the screen. Task G involved putting nuts into a can and withdrawing them one by one; the participants responded after each withdrawal as to whether the can still contained any nuts or was empty. Task H involved placing candy inside the box with a number of fish drawn on the lid; the box was then hidden and brought out again along with another box marked with one more or one less fish on the lid, and participants had to choose which box contained the candy. As can be seen from the associated graphs, the Pirahã did well only when the number of objects involved was smaller than four, and failed on harder tasks that included as few as three objects. Gordon concluded that “the split between exact enumeration ability for set sizes smaller than three and analog estimation for larger set sizes” among the Pirahã parallels findings from laboratory experiments with “prelinguistic infants, monkeys, birds, and rodents”. English-speaking adult controls had no problem with any of these experimental conditions, performing with a much higher degree of accuracy than the Pirahã speakers. Thus, Gordon’s results appear to confirm Dan Everett’s claim that the lack of number words in Pirahã goes beyond a linguistic deficiency: the Pirahã seem to lack the necessary cognitive mechanisms that would allow them to appreciate numerosities beyond two or three. According to Gordon, “numerical cognition is clearly affected by the lack of a counting system in the language”. But is this claim necessarily true?
A more recent study conducted by a team of researchers led by MIT professor of brain and cognitive sciences Edward Gibson*, challenged Gordon’s conclusions. These scholars found that the Pirahã succeeded in simple one-to-one matching task, suggesting that the concept of exact quantity—that is the idea that adding or subtracting one object makes a difference—does not depend on linguistic knowledge. Even more interestingly, Gibson and his team showed that when it comes to the counting, English speakers—even MIT students—can appear to be as innumerate as Pirahã speakers if they are not allowed to use verbal counting. In order to do prevent English speaking subjects from counting silently in their head, the experimenters used the so-called “verbal shadowing technique”: the subjects had to repeat a meaningful sequence of words while performing the experimental counting task at hand. In this experiment, twenty participants from MIT and the surrounding community, ranging in age from 18 to 50, had to perform five matching tasks like the ones used in Gordon’s experiment with the Pirahã. The tasks required the participants to observe some quantity of thread spools and to put out a line of uninflated balloons exactly matching the quantity of spools that they saw, which varied from 4 to 12. The task were:
- a one-to-one matching task: the spools evenly placed from the participant’s left to their right
- an uneven matching task: the spools were broken into smaller groups of one to four
- an orthogonal matching task: the line of spools ran from close to the participant to further away, rather than from left to right
- a hidden matching task: the line was hidden by the experimenter after the spools were placed by placing a manila folder in front of the spools
- a nuts-in-a-can task: the experimenter placed spools one by one into an opaque cup
On each trial, the experimenter would begin by starting the audio sequence that the participant listened to over headphones. The participants had to repeat the text that they heard while performing the matching tasks. It was found that for the English speakers, as for the Piraha, the one-to-one matching and uneven matching tasks were easiest, followed by the orthogonal and hidden matching tasks, while the nuts-in-a-can task was by far the hardest. While performance in one-to-one matching tasks was largely preserved without the use of verbal counting, performance on more memory-intensive tasks was dramatically impaired. Thus, occupying the verbal resources of speakers of a language that has numbers produced a pattern of data remarkably similar to the data of speakers of a language without numbers, leading the researchers to deduce that memory for exact quantities, but not one-to-one matching, is dependent on language.
Gibson and his team concluded that all human beings share a variety of core numerical capacities (which is probably shared with many other species as well). But both learning and using the ability to remember exact quantities larger than three or four appears to depend crucially on verbal mechanisms. Thus, languages that contain recursive count lists allow their speakers to transcend their core numerical capacities.
* Other members of the team include Michael Frank, a graduate student in Gibson’s lab, Evelina Fedorenko, a postdoctoral associate at the McGovern Institute for Brain Research at MIT, and Dan Everett.
Frank, M. C., Everett, D. L., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition 108(3): 819-824.
Nevins, A., Pesetsky, D., Rodrigues, C. (2009). Pirahã Exceptionality: A Reassessment. Language 85(2), 355-404