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 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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