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Numerical Cognition Without Words: Evidence From Amazonia

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Numerical Cognition Without Words: Evidence From Amazonia

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Members of the Pirahã tribe use a “one-two-many” system of counting. I ask whether speakers of this innumerate language can appreciate larger numerosities without the benefit of words to encode them. This addresses the classic Whorfian question about whether language can determine thought. Results of numerical tasks with varying cognitive demands show that numerical cognition is clearly affected by the lack of a counting system in the language. Performance with quantities greater than three was remarkably poor, but showed a constant coefficient of variation, which is suggestive of an analog estimation process.
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Numerical Cognition Without
Words: Evidence from Amazonia
Peter Gordon
Members of the Piraha
˜
tribe use a ‘one-two-many’ system of counting. I ask
whether speakers of this innumerate language can appreciate larger
numerosities without the benefit of words to encode them. This addresses
the classic Whorfian question about whether language can determine
thought. Results of numerical tasks with varying cognitive demands show
that numerical cognition is clearly affected by the lack of a counting system
in the language. Performance with quantities greater than three was
remarkably poor, but showed a constant coefficient of variation, which is
suggestive of an analog estimation process.
Is it possible that there are some concepts
that we cannot entertain because of the
language that we speak? At issue here is
the strongest version of Benjamin Lee
Whorf_s hypothesis that language can deter-
mine the nature and content of thought. The
strong version of Whorf_s hypothesis goes
beyond the weaker claim that linguistic
structure simply influences the way that we
think about things in our everyday encoun-
ters. For example, recent studies suggest that
language might affect how people mentally
encode spatial relations (1–3), and how they
conceive of the nature of individual objects
and their material substances (4). However,
none of these studies suggest that linguistic
structure prevents us from entertaining the
concepts that are available to speakers of
alternative linguistic systems.
The question of whether linguistic de-
terminism exists in the stronger sense has
two parts. The first is whether languages
can be incommensurate: Are there terms
that exist in one language that cannot be
translated into another? The second is
whether the lack of such translation pre-
cludes the speakers of one language from
entertaining concepts that are encoded by
the words or grammar of the other language.
For many years, the answer to both ques-
tions appeared to be negative. Although
languages might have different ways in
which situations are habitually described, it
has generally been accepted that there
wouldalwaysbesomewayinwhichone
could capture the equivalent meaning in any
other language (5). Of course, when speak-
ing of translatable concepts, we do not
mean terms like Bmolecule[ or Bquark,[
which would not exist in a culture without
advanced scientific institutions. Failure to
know what molecules or quarks are does not
signal an inability to understand the English
language—surely people were still speaking
English before such terms were introduced.
On the other hand, one would question
someone_s command of English if they did
not understand the basic vocabulary and
grammar.
Words that indicate numerical quantities
are clearly among the basic vocabulary of a
language like English. But not all languages
contain fully elaborated counting systems.
Although no language has been recorded that
completely lacks number words, there is a
considerable range of counting systems that
exists across cultures. Some cultures use a
finite number of body parts to count 20 or 30
body tags (6). Many cultures use particular
body parts like fingers as a recursive base for
the count system as in our 10-based system.
Finally, there are cultures that base their
counting systems on a small number between
2 and 4. Sometimes, the use of a small-
number base is recursive and potentially
infinite. For example, it is claimed that the
Gumulgal South Sea Islanders counted with a
recursive binary system: 1, 2, 2_1, 2_2, 2_2_1,
and so on (6).
The counting system that differs perhaps
most from our own is the Bone-two-many[
system, where quantities beyond two are not
countedbutaresimplyreferredtoas
Bmany.[ If a culture is limited to such a
counting system, is it possible for its
members to perceive or conceptualize quan-
tities beyond the limited sets picked out by
the counting sequence, or to make what we
consider to be quite trivial distinctions such
as that between four versus five objects?
The Pirah, are such a culture. They live
along the banks of the Maici River in the
Lowland Amazonia region of Brazil. They
maintain a predominantly hunter-gatherer
existence and reject assimilation into main-
stream Brazilian culture. Almost completely
monolingual in their own language, they
have a population of less than 200 living in
small villages of 10 to 20 people. They have
only limited exchanges with outsiders, using
primitive pidgin systems for communicating
in trading goods without monetary exchange
and without the use of Portuguese count
words. The Pirah, counting system consists
of the words: BhFi[ (falling tone 0Bone[) and
Bho<[ (rising tone 0Btwo[). Larger quantities
are designated as Bbaagi[ or Baibai[ (0Bmany[).
I was able to take three field trips,
ranging from 1 week to 2 months, living
with the Pirah, along with Daniel Everett
and Keren Everett, two linguists who have
lived and worked with the tribe for over 20
years and are completely familiar with their
language and cultural practices. Observations
were informed by their background of con-
tinuous and extensive immersion in the Pirah,
culture. During my visits, I became interested
in the counting system of the Pirah, that I had
heard about and wanted to examine whether
they really did have only two numbers and
how this would affect their ability to perceive
numerosities that extended beyond the limited
count sequence.
Year 1: Initial observations. On my first
week-long trip to the two most up-river
Maici villages, I began with informal obser-
vations of the Pirah, use of the number
words for one and two. I was also interested
in the possibility that the one-two-many
system might actually be a recursive base-2
system, that their limited number words
might be supplemented by more extensive
finger counting, or that there might be taboos
associated with counting certain kinds of
objects as suggested by Zaslavsky in her
studies of African counting systems (7, 8).
Keren Everett developed some simple tasks
to see if our two Pirah, informants could
refer to numerosities of arrays of objects
using Pirah, terms and any finger counting
system they might have. Instructions and
interactions with participants were in the
Pirah, language. When it was necessary to
refer to the numerosity of an array, Keren
Department of Biobehavioral Sciences, Columbia
University, 525 West 120th Street, New York, NY
10027, USA. E-mail: pgordon@tc.columbia.edu
Table 1. Use of fingers and number words by
Piraha
˜
participant. The arrow (Y) indicates a shift
from one quantity to the next.
No. of objects
Number
word used
No. of fingers
1 ho
´
i(0 1)
2 hoı
´
(0 2) 2
aibaagi (0 many)
3 hoı
´
(0 2) 3
4 hoı
´
(0 2) 5 Y 3
aibai (0 many)
5 aibaagi (0 many) 5
6 aibaagi (0 many) 6 Y 7
7 ho
´
i(0 1)* 1
aibaagi (0 many) 5 Y 8
8 5 Y 8 Y 10
9 aibaagi (0 many) 5 Y 10
10 5
*This use of ‘‘one’’ might have been a reference to adding
one rather than to the whole set of objects.
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Everett used the Portuguese number words
embedded in Pirah, dialogue. Such terms are
understood by the Pirah, to be the language
of Brazilians, but their meaning is not
understood. In addition to this short session,
during the first year trip, I continuously took
opportunities to probe for counting abilities
in everyday situations.
The outcome of these informal studies re-
vealed the following: (i) There was no re-
cursive use of the count system—the Pirah,
never used the count words in combinations
like BhFi-ho<[ to designate larger quantities.
(ii) Fingers were used to supplement oral
enumeration, but this was highly inaccurate
even for small numbers less than five. In
addition, BhFi[ and Bho<,[ the words for
Bone[ and Btwo,[ were not always used to
denote those quantities. Whereas the word for
Btwo[ always denoted a larger quantity than
the word for Bone[ (when used in the same
context), the word for Bone[ was sometimes
used to denote just a small quantity such as
two or three or sometimes more. An example
of the use of counting words and finger
counting is given in Table 1 in one of the
informal sessions with an informant who
appeared to be in his 50s. Videotaped ex-
tracts from the session are included in the
supporting online materials (movie S1).
The interpretation of these observations is
limited by their informal nature and small
sample size. However, the observations are
supplemented with 20 years of observation
by the Everetts as trained linguists in their
analysis of the Pirah, language. One partic-
ularly interesting finding is that BhFi[
appears to designate Broughly one[—or a
small quantity whose prototype is one. Most
of the time, in the enumeration task, BhFi[
referred to one, but not always. An analogy
might be when we ask for Ba couple of Xs[
in English, where the prototypical quantity is
two, but we are not upset if we are given
three or four objects. However, we surely
would be upset if given only one object,
because the designation of a single object
has a privileged status in our language.
There is no concept of Broughly one[ in a
true integer system. Even the informal use
of the indefinite article Ba X[ strictly
requires a singular reference. In Pirah,,
BhFi[ can also mean Bsmall,[ which con-
trasts with Bogii[ (0 big), suggesting that the
distinction between discrete and continuous
quantification is quite fuzzy in the Pirah,
language.
Year 2: Experiments in nonverbal numer-
ical reasoning. On my second visit to the
Pirah, villages for a 2-month period, I
developed a more systematic set of proce-
dures for evaluating the numerical compe-
tence of members of the tribe. The
experiments were designed to require some
combination of cognitive skills such as the
need for memory, speed of encoding, and
mental-spatial transformations. This would
reveal the extent to which such task demands
interact with numerical ability, such as it is.
Details of the methods are available on
Science Online (9). There were seven partic-
ipants, who included all six adult males from
two villages and one female. Most of the data
were collected on four of the men who were
consistently available for participation. The
tasks were devised to use objects that were
available and familiar to the participants
(sticks, nuts, and batteries). The results of
the tasks, along with schematic diagrams, are
presented in Fig. 1. These are roughly
ordered in terms of increasing cognitive
demand. Any estimation of a person_s
numerical competence will always be con-
founded with performance factors of the task.
Because this is unavoidable, it makes sense to
explore how performance is affected by a
range of increasingly demanding tasks.
In the matching tasks (A, B, C, D, and F),
I sat across from the participant and with a
stick dividing my side from theirs, I
presented an array of objects on my side of
the stick (below the line in the figures) and
they responded by placing a linear array of
AA batteries (5.0 cm by 1.4 cm) on their
side of the table (above the line). The
matching task provides a kind of concrete
substitute for counting. It shares the element
ABC
0.00
0.25
0.50
0.75
1.00
tc
erroC .porP
12345678910
Target
1-1 Line Match
0.00
0.25
0.50
0.75
1.00
t
c
er
r
oC .p
o
r
P
12345678910
Target
Cluster Line Match
0.00
0.25
0.50
0.75
1.00
tce
r
roC
.porP
12345678910
Target
Orthogonal Line Match
DE F
0.00
0.25
0.50
0.75
1.00
tcerroC .porP
12345678910
Target
Uneven Line Match
0.00
0.25
0.50
0.75
1.00
t
cerro
C
.p
o
r
P
12345678910
Target
Line Draw Copy
0.00
0.25
0.50
0.75
1.00
tc
er
roC
.p
o
r
P
12345678910
Target
Brief Presentation
GH
0.00
0.25
0.50
0.75
1.00
tcerroC .porP
12345678910
Target
Nuts-in-Can Task
0.00
0.25
0.50
0.75
1.00
tcerroC .porP
Target
Candy-in-Box Task
1v2 2v3 3v4 4v5 5v6 6v7
Fig. 1. Results of number tasks with Piraha
˜
villagers (n 0 7). Rectangles indicate AA batteries (5.0 cm
by 1.4 cm), and circles indicate ground nuts. Center line indicates a stick between the author’s
example array (below the line) and the participant’s attempt to ‘‘make it the same (above the line).
Tasks A through D required the participant to match the lower array presented by the author using a
line of batteries; task E was similar, but involved the unfamiliar task of copying lines drawn on paper;
task F was a matching task where the participant saw the numerical display for only about 1 s before
it was hidden behind a screen; task G involved putting nuts into a can and withdrawing them one by
one; (participants responded after each withdrawal as to whether the can still contained nuts or was
empty); task H involved placing candy inside a box with a number of fish drawn on the lid (this was
then hidden and brought out again with another box with one more or one less fish on the lid, and
participants had to choose which box contained the candy).
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of placing tokens in one-to-one correspon-
dence with individuals in a to-be-counted
group. The first matching tasks began with
simple linear arrays of batteries. This prog-
ressed to clusters of nuts matched to the
battery line, orthogonal matching of battery
lines, matching of battery lines that were
unevenly spaced, and copying lines on a
drawing. In all of these matching experi-
ments, participants responded with relatively
good accuracy with up to 2 or 3 items, but
performance deteriorated considerably be-
yond that up to 8 to 10 items. In the first
simple linear matching task A, performance
hovered around 75% up to the largest
quantities. Matching tasks with greater
cognitive demands required mental transpo-
sition of the sample array to the match array
without benefit of tagging for numerical
quantity. Performance dropped precipitously
to 0% for the larger target set sizes in these
tasks. One exception was task D with
unevenly spaced objects. Although this was
designed to be a difficult task, participants
showed an anomalous superiority for large
numerosities over small. Performance initial-
ly deteriorated with increased set size up to 6
items, then shot up to near perfect perfor-
mance for set size 7 through 10. A likely
interpretation of this result was that the
uneven spacing for larger set sizes promoted
recoding of arrays into smaller configurations
of two or three items. This allowed partic-
ipants to use a chunking strategy of treating
each of the subgroups as a matching group.
When time constraints were introduced
in task F (exposing the array for only 1 s),
performance was drastically affected and
there was a clear correlation between set
size and accuracy beginning at set size 3. A
line-drawing task (E) was highly affected by
set size, being one of the worst perform-
ances of all. Not only do the Pirah, not
count, but they also do not draw. Producing
simple straight lines was accomplished only
with great effort and concentration, accom-
panied by heavy sighs and groans. The final
two tasks (G and H) required participants to
keep track of a numerical quantity through
visual displacement. In one case, they were
first allowed to inspect an array of nuts for
about 8 s. The nuts were placed in a can, and
then withdrawn one at a time. Participants
were required to say, after each withdrawal,
if there were still any nuts left in the can or
if it was empty. Performance was predict-
ably strongly affected by set size from the
very smallest quantities. The final task
involved hiding candy in a box, which had
a picture of some number of fish on the lid.
The box was then hidden behind the
author_s back, and two cases were revealed,
the original with the candy, and another
with one more or one less fish on the lid. For
quite small comparisons such as three versus
four, performance rarely went over 50%
chance responding.
There is a growing consensus in the field of
numerical cognition that primitive numerical
abilities are of two kinds: First, there is the
ability to enumerate accurately small quantities
up to about three items, with only minimal
processing requirements (10–16). I originally
termed this ability Bparallel individuation[
(17, 18), referring to how many items one
can encode as discrete unique individuals at
the same time in memory. Without overt
counting, humans and other animals possess
an analog procedure whereby numerical
quantities can be estimated with a limited
degree of accuracy (11, 19–26). Many
researchers believe that large-number estima-
tion, although based on individuated elements,
is coalesced into a continuous analog format
for mental representation. For example, the
discrete elements of a large number array
might be represented as a continuous length of
a line, where a longer line inexactly represents
a larger numerosity.
When people use this analog estimation
procedure, the variability of their estimates
tends to increase as the target set size
increases. The ratio of average error to
target set size is known as Weber_s fraction
and can be indexed by a measure known as
the coefficient of variation—the standard
deviation of the estimates divided by set size
(23). Although performance by the Pirah,
on the present tasks was quite poor for set
sizes above two or three, it was not random.
Figure 2 shows the mean response values
mapped against the target values for all
participants in the simple matching tasks A,
B, C, and F. The top graph shows that mean
responses and target values are almost
identical. This means that the Pirah, partic-
ipants were trying hard to get the answers
correct, and they clearly understood the
tasks. The lower graph in Fig. 2 shows that
the standard deviation of the estimates
increases in proportion to the set size,
resulting in a constant coefficient of varia-
tion of about 0.15 after set size three, as
predicted by the dual model of mental
enumeration. This value for the coefficient
of variation is about the same as one finds in
college students engaged in numerical esti-
mation tasks (23). Data for individual tasks
and individual participants were consistent
with the averaged trends shown in Fig. 2.
Graphs are available in the supporting
online materials (figs. S2 and S3).
The results of these studies show that the
Pirah,_s impoverished counting system lim-
its their ability to enumerate exact quantities
when set sizes exceed two or three items.
For tasks that required additional cognitive
processing, performance deteriorated even
on set sizes smaller than three. Participants
showed evidence of using analog magnitude
estimation and, in some cases, they took
advantage of spatial chunking to decrease
the cognitive demands of larger set sizes.
This split between exact enumeration ability
for set sizes smaller than three and analog
estimation for larger set sizes parallels
findings from laboratory experiments with
adults who are prevented from explicit
counting; studies of numerical abilities in
prelinguistic infants, monkeys, birds, and
rodents; and in recent studies using brain-
imaging techniques (11, 23–30).
The analog estimation abilities exhibited
by the Pirah, are a kind of numerical
competence that appears to be immune to
numerical language deprivation. But because
lower animals also exhibit such abilities,
robustness in the absence of language is
already established. The present experiments
allow us to ask whether humans who are not
exposed to a number system can represent
exact quantities for medium-sized sets of
four or five. The answer appears to be
negative. The Pirah, inherit just the abilities
to exactly enumerate small sets of less than
three items if processing factors are not
unduly taxing (31).
In evaluating the case for linguistic deter-
minism, I suggest that the Pirah, language is
incommensurate with languages that have
counting systems that enable exact enumer-
ation. Of particular interest is the fact that the
Pirah, have no privileged name for the singular
quantity. Instead, BhFi[ meant Broughly one[
or Bsmall,[ which precludes any precise trans-
lation of exact numerical terms. The present
study represents a rare and perhaps unique
case for strong linguistic determinism. The
A
0
1
2
3
4
5
6
7
8
9
10
123456789
Target
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
B
0.00
0.15
0.30
123456789
Target
Mean
DS
VC
Mean
SD
Fig. 2. (A) Mean accuracy and standard
deviation of responses in matching tasks and
(B) coefficient of variation. Figures for individ-
ual tasks and individual participants are avail-
able in the supporting online materials.
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15 OCTOBER 2004 VOL 306 SCIENCE www.sciencemag.org
498
study also provides a window into how the
possibly innate distinction (26) between quan-
tifying small versus large sets of objects is
relatively unelaborated in a life without num-
ber words to capture those exact magnitudes
(32).
References and Notes
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5. R. W. Brown, E. H. Lenneberg, J. Abnorm. Soc.
Psychol. 49, 454 (1954).
6. K. Menninger, Number Words and Number Symbols:
A Cultural History of Numbers (MIT Press, Cambridge,
MA, 1969).
7. C. Zaslavsky, Africa Counts: Number and Pattern in
African Culture (Prindle, Weber, and Schmidt, Boston,
1973).
8. R. Gelman, C. R. Gallistel, The Child’s Understanding
of Number (Harvard Univ. Press, Cambridge, MA,
1978).
9. Materials and methods are available as supporting
material on Science Online.
10. S. Carey, Mind Lang. 16, 37 (2001).
11. L. Feigenson, S. Carey, M. Hauser, Psychol. Sci. 13,
150 (2002).
12. B. J. Scholl, Cognition 80, 1 (2001).
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17. P. Gordon, paper presented at the biennial meeting
of the Society for Research in Child Development,
New Orleans, LA, 25 to 28 March 1993.
18. P. Gordon, paper presented at the European Society
for Philosophy and Psychology, Paris, France, 1 to 4
September 1994.
19. W. H. Meck, R. M. Church, J. Exp. Psychol. Anim.
Behav. Processes 9, 320 (1983).
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Psychon. Bull. Rev. 8, 698 (2001).
22. C. R. Gallistel, The Organization of Learning (MIT
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Psychon. Bull. Rev. 8, 698 (2001).
25. S. Dehaene, The Number Sense (Oxford Univ. Press,
New York, 1997).
26. B. Butterworth, What Counts (Simon & Schuster,
New York, 1999).
27. J. S. Lipton, E. S. Spelke, Psychol. Sci. 14,396
(2003).
28. M. D. Hauser, F. Tsao, P. Garcia, E. S. Spelke, Proc. R.
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Science 284, 970 (1999).
30. J. R. Platt, D. M. Johnson, Learn. Motiv. 2, 386 (1971).
31. Cordes et al.(24) suggest that analog repre-
sentations exist even for n 0 2, because subjects
made errors on a task in which counting was sup-
pressed during rapid button pressing. However,
errors in this range also occurred when subjects
counted and might have been the result of per-
severation errors rather than reflecting numerical
representations.
32. One can safely rule out that the Piraha
˜
are mentally
retarded. Their hunting, spatial, categorization, and
linguistic skills are remarkable, and they show no
clinical signs of retardation.
33. I thank D. Everett and K. Everett for making this
research possible; SIL in Porto Velho and E. Ramos
for logistical support; and S. Carey, L. Feigenson,
D. Everett, C. Tamis-LeMonda, M. Miozzo, G. Marcus,
F. Xu, and K. Adolph for comments on the paper.
Supporting Online Material
www.sciencemag.org/cgi/content/full/1094492/DC1
Methods
SOM Text
Figs. S1 to S3
Movies S1 and S2
References
9 December 2003; accepted 29 July 2004
Published online 19 August 2004;
10.1126/science.1094492
Include this information when citing this paper.
Exact and Approximate Arithmetic
in an Amazonian Indigene Group
Pierre Pica,
1
Cathy Lemer,
2
Ve
´
ronique Izard,
2
Stanislas Dehaene
2
*
Is calculation possible without language? Or is the human ability for
arithmetic dependent on the language faculty? To clarify the relation
between language and arithmetic, we studied numerical cognition in speakers
of Munduruku
´
, an Amazonian language with a very small lexicon of number
words. Although the Munduruku
´
lack words for numbers beyond 5, they are
able to compare and add large approximate numbers that are far beyond their
naming range. However, they fail in exact arithmetic with numbers larger
than 4 or 5. Our results imply a distinction between a nonverbal system of
number approximation and a language-based counting system for exact
number and arithmetic.
All science requires mathematics. The
knowledge of mathematical things is
almost innate in usI . This is the
easiest of sciences, a fact which is
obvious in that no one_s brain rejects
it; for laymen and people who are
utterly illiterate know how to count
and reckon.
Roger Bacon (1214–1294),
English philosopher and scientist
Where does arithmetic come from? For some
theorists, the origins of human competence in
arithmetic lie in the recursive character of the
language faculty (1). Chomsky, for instance,
stated that Bwe might think of the human
number faculty as essentially an Fabstraction_
from human language, preserving the mech-
anisms of discrete infinity and eliminating
the other special features of language[ (2).
Other theorists believe that language is not
essential—that humans, like many animals,
have a nonverbal Bnumber sense[ (3), an
evolutionarily ancient capacity to process
approximate numbers without symbols or
language (4–6) that provides the conceptual
foundation of arithmetic. A third class of
theories, while acknowledging the existence
of nonverbal representations of numbers,
postulates that arithmetic competence is
deeply transformed once children acquire a
system of number symbols (7–9). Language
would play an essential role in linking up the
various nonverbal representations to create a
concept of large exact number (10–12).
To elucidate the relations between language
and arithmetic, it is necessary to study numer-
ical competence in situations in which the
language of numbers is either absent or
reduced. In many animal species, as well as
in young infants before they acquire number
words, behavioral and neurophysiological
experiments have revealed the rudiments of
arithmetic (6, 13–16). Infants and animals
appear to represent only the first three
numbers exactly. Beyond this range, they
can approximate Bnumerosity,[ with a fuzzi-
ness that increases linearly with the size of the
numbers involved (Weber_s law). This finding
and the results of other neuroimaging and
neuropsychological experiments have yielded
a tentative reconciliation of the above theo-
ries: Exact arithmetic would require language,
whereas approximation would not (12, 17–21).
This conclusion, however, has been chal-
lenged by a few case studies of adult brain-
lesioned or autistic patients in whom language
dysfunction did not abolish exact arithmetic;
such a finding suggests that in some rare cases,
even complex calculation may be performed
without words (22).
In the final analysis, the debate cannot be
settled by studying people who are raised in
a culture teeming with spoken and written
symbols for numbers. What is needed is a
language deprivation experiment, in which
neurologically normal adults would be raised
1
Unite
´
Mixte de Recherche 7023 ‘Formal Structures
of Language,’ CNRS and Paris VIII University, Paris,
France.
2
Unite
´
INSERM 562 ‘Cognitive Neuroimag-
ing,’ Service Hospitalier Fre
´
de
´
ric Joliot, CEA/DSV,
91401 Orsay Cedex, France.
*To whom correspondence should be addressed.
E-mail: dehaene@shfj.cea.fr
R EPORTS
www.sciencemag.org SCIENCE VOL 306 15 OCTOBER 2004
499
... Other research emphasizes that language is helpful in supporting number. [40,41] administered non-linguistic, exact number tasks to the Pirahã, an Amazonian group whose language has no exact number words. While subjects succeeded on several tasks, they had trouble in tasks involving larger sets and memorization. ...
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... This numeral system is used by a tribe, Pirahã, living in Amazonia nowadays. A study published in Science in 2004 (see [23]) describes that these people use an extremely simple numeral system for counting: one, two, many. For Pirahã, all quantities larger than two are just 'many' and such operations as 2 + 2 and 2 + 1 give the same result, i.e., 'many'. ...
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... Particularly, we estimate precisely, without counting, a small set of items [12]. The above-mentioned mechanism is present across different human societies [13] as it has been evolutionary conserved within a wide variety of other species-from bees to nonhuman primates-highlighting its importance for survival and its crucial role for social behavior [14][15][16]. There is a broad agreement that representations of numerosity are analog and multi-sensory: embedded within a sequences of sounds, in the cardinality of figures or in a series of actions [17]. ...
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