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HOME > PAST ISSUE > March-April 2005 > Article Detail

SCIENCE OBSERVER

Can You Count Your Chickens?

Roger Harris

Imagine life without numbers. Science as we know it would be impossible and, indeed, so would modern civilization. Yet, some societies get along just fine without numbers greater than four or five. Two studies of Amazonian rainforest tribes reported simultaneously in the journal Science (October 15) used this paucity to shed light on how people perceive, learn and reason with numbers.

In Missao Velha, Brazil...Click to Enlarge Image

In one of the investigations, a French team of cognitive neuroscientists led by Pierre Pica from Paris University studied speakers of Mundurukú in central Brazil. Mundurukú lacks vocabulary for numbers beyond 5, so it's a natural experiment for investigating the process of counting—associating real things with abstract symbols such as words.

But how is anyone able to count at all? To answer this question, team member Stanislas Dehaene developed a neuronal-network model of number processing, which is built on behavioral and brain-imaging studies. He hypothesizes that human beings have a "number sense," an evolutionarily ancient cerebral system that enables people to approximate quantities. This system is capable only of simple computations, but humans have tweaked it by inventing cultural "tools" such as number symbols and counting routines. These confer  the ability to perform accurate calculations. Because the Mundurukú have so few number words, they're ideal to test this hypothesis. To do so, the French scientists devised a series of simple experiments based on presenting subjects with various numbers of dots on a solar-powered laptop computer screen.

The first batch of tests showed that Mundurukú people inconsistently use even the number words they do have. Given four dots, a person might use the word for 3, 4 or 5. Words for numbers above 2 could refer to a range of quantities: The Mundurukú word for 5, synonymous with "a handful," was applied just as readily to 6, 7, 8 or 9 as it was to 5. Speakers of numerate languages are usually exact in their use of number words in this range.

It seems that the Mundurukú use a particular word based on an approximation rather than an exact count. This may be akin to certain designations in English. "I had a couple of drinks," could mean you had two drinks, or three or four drinks. Likewise, words such as "several" and "few" refer to an inexact quantity.

The team next evaluated how well the Mundurukú estimate large numbers. The Indians' skill at comparing large groups of dots was on a par with Westerners. Dehaene considers this the most important discovery: "I was stupefied to see that these isolated, uneducated people ... could assess the numerosity of a set of 50 dots with exactly the same precision as educated French speakers!"

A final round of tasks judged the Mundurukú's capacity for exact subtraction. Their scores were consistent with the "number sense" model's prediction that without words people will represent numbers inexactly: Performance decreased as the initial number increased, especially above 4 (for example, 5 minus 2). Thus the Mundurukú apparently used approximation to subtract when the French controls would use exact calculation.

About 700 miles west of the French team's site, behavioral scientist Peter Gordon at Columbia University had also surveyed numerical ability among Brazilian Indians during the early 1990s. He worked with members of the Pirahã tribe, whose entire lexicon for precise numbers consists of the words for 1 and 2. To indicate greater quantities, the Pirahã use "aibaagi," which translates to "many." To find out whether this limited vocabulary hindered arithmetic ability, Gordon asked the Indians to match numbers of various objects such as AA batteries, drawn lines, nuts and candy. For more than three objects, the participants' accuracy at matching assemblages having the same number dropped significantly. Given more complex tasks, performance was poor even on trials involving numbers below 3—perhaps because, Gordon says, the problems "required additional cognitive processing."

Despite their different methods, both teams came to more or less the same conclusion: Accurately counting above 4 or 5 requires precise number names. They also agree that estimating and comparing large quantities is independent of the ability to count exactly. Remarkably, this disassociation of accurate counting versus approximation corresponds to developmental progress: Children have difficulty grasping numbers greater than two until the age of three or four. All this research fits the idea that people have two mental systems for dealing with numbers: One for small numbers, less than 5 or so, and the other for large numbers, enabling people to approximate.

If mental faculties of human beings are the same, then why do cultures differ in their aptitude for handling numbers? For example, the French control group performed exact subtraction more accurately than the Mundurukú speakers did. Why?

On this point, the French workers and Gordon disagree. Pica believes that differences are explained by "performance," which depends on having a counting system. In the case of the Indians, lack of a counting system explains their lower scores in the subtraction tests. He characterizes Gordon as explaining differences by "competence" (innate faculty to count) and says, "Gordon asserts … that, cognitively speaking, the Indians are different—a position I [Pica] strongly deny." Stanislas Dehaene feels this polarization is excessive and has a different view to Pica's: that the Mundurukú have not invented the "tool" of counting and so do not routinely count.

Gordon for his part asserts that the differences are based on cognitive disparities that arise from language. In his view, the Pirahã's dearth of number words hinders their ability to conceptualize beyond 2 or 3: Without language, people cannot count. He finds "the whole competence-performance angle puzzling" because the terminology was developed with respect to language and is not readily applicable to number studies—a view ostensibly supported by neuropsychologist Marc D. Hauser of Harvard University, who says "the competence-performance distinction has been a notoriously difficult one to establish." In linguistics, competence refers to our intrinsic knowledge of grammar, whereas performance is the often-imperfect manifestation of grammar as speech.

Maybe the debate is purely semantic: Swap Gordon's "language" for Pica's "counting system," and the division between competence and performance disappears. Given this perspective, perhaps it's not valid to frame cultural differences in terms of competence versus performance.

There's a curious footnote to this story. By November, the popular scientific press was abuzz with speculation on how Gordon's results shed light on linguist Benjamin Lee Whorf's controversial hypothesis that language determines the nature and content of thought. These conjectures remain unresolved, but they beg the question of why Pica's work was ignored. Cognitive scientist Rochel Gelman at Rutgers University suggests this disparity reflects a Lockean-Cartesian dichotomy. The writings of John Locke, who was the first to assert the empiricist view that all knowledge comes from experience, have influenced scientific ideas among English speakers (including the Founding Fathers). According to Gelman, English speakers have a "huge bias" against French science, perhaps because of its origins in René Descartes' mind-matter dualism ("I think, therefore I am"). Although it may be real, the bias is apparently baseless: Dehaene says that French scientists generally accept the empiricist view and state "some of the strongest anti-dualist positions." His explanation for the lack of media coverage: Gordon's paper was published online, with much fanfare, six weeks before the appearance of the print magazine.

Yet, the Lockean-Cartesian factionalism fits neatly with Pica and Gordon's differences in their explanations of counting among Amazonian Indians. Is it the root cause of their supposed differences? Perhaps it would be best to leave that argument to the philosophers.— Roger Harris


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