ملاحظات
تمهيد
(1)
For some accounts of shipwrecked sailors
surviving with indigenous cultures, see Alvar Núñez
Cabeza de Vaca, The Shipwrecked
Men (London: Penguin Books, 2007).
(2)
See, for example, Brian Cotterrell and
Johan Kamminga, Mechanics of
Pre-Industrial Technology (Cambridge:
Cambridge University Press, 1990).
(3)
For more on the cultural ratchet, see
Claudio Tennie, Josep Call, and Michael Tomasello,
“Ratcheting Up the Ratchet: On the Evolution of
Cumulative Culture,” Philosophical Transactions of the Royal Society
B 364 (2009): 2405–2415, as well as
Michael Tomasello, The Cultural
Origins of Human Cognition (Cambridge,
MA: Harvard University Press, 2009).
(4)
For discussion of this Inuit case, and for
elaboration of the notion of culturally stored
knowledge, see Robert Boyd, Peter Richerson, and Joseph
Henrich, “The Cultural Niche: Why Social Learning Is
Essential for Human Adaptation,” Proceedings of the National Academy of Sciences
USA 108 (2011): 10918–10925. For more on
the evolution of cultures, see, for example, Peter
Richerson and Morten Christiansen, eds., Cultural Evolution: Society,
Technology, Language, and Religion.
Strüngmann Forum Reports, volume 12 (Cambridge, MA: MIT
Press, 2013).
الجزء الأول: تغلغل الأعداد في الخبرة البشرية
الفصل الأول: الأعداد منسوجة في حاضِرنا
(1)
For more on the perception of time
among the Aymara, see Rafael Núñez and Eve Sweetser,
“With the Future behind Them: Convergent Evidence
from Aymara Language and Gesture in the
Crosslinguistic Comparison of Spatial Construals of
Time,” Cognitive
Science 30 (2006):
401–450.
(2)
Thaayorre temporal perception is
analyzed in Lera Boroditsky and Alice Gaby,
“Remembrances of Times East: Absolute Spatial
Representations of Time in an Australian Aboriginal
Community,” Psychological
Science 21 (2010):
1621–1639.
(3)
In a related vein, it is worth noting
that the duration of the earth’s rotation (whether
sidereal or with respect to the sun) is not
absolute. For instance, prior to the moon-creating
collision of a planetesimal with the earth billions
of years ago, the earth’s solar day lasted only
about six hours. Even now days are gradually
increasing in duration as the rotation of the earth
slows bit by bit due to tidal friction, and
furthermore solar days vary slightly depending on
the earth’s orbital position relative to the sun.
For more on this topic, see, for instance, Jo Ellen
Barnett, Time’s Pendulum:
From Sundials to Atomic Clocks, the Fascinating
History of Timekeeping and How Our Discoveries
Changed the World (San Diego:
Harcourt Brace, 1999).
(4)
It is also the result of the
development of associated mechanisms used to keep
track of time, from sundials to smart phones.
Interestingly, this development reflects the
increasingly abstract nature of time-keeping. Where
once such mechanisms, like sundials and water
clocks, were used to track the diurnal cycle, they
eventually came to track units of time that are
independent of celestial patterns. This transition
stems in part from the development of weight-based
clocks (particularly pendulum clocks) and
spring-based time pieces, which allowed for more
accurate measurement of time than any celestial
methods available. Such accurate time measurement
enabled, among other major innovations, more precise
longitude measurement and navigation. See the
fascinating discussion in Barnett, Time’s
Pendulum.
(5)
There are many excellent books on
human evolution and paleoarchaeology. For one
recent exemplar, see Martin Meredith, Born in Africa: The Quest for
the Origins of Human Life (New
York: Public Aff airs,
2012).
(6)
The claims regarding
australopithecines are based on the famous work
of the Leakeys, notably in Mary Leakey and John
Harris, Laetoli:
A Pliocene Site in Northern
Tanzania (New York: Oxford
University Press, 1979), as well as Mary Leakey
and Richard Hay, “Pliocene Footprints in the
Laetolil Beds at Laetoli, Northern Tanzania,”
Nature
278 (1979): 317–323. See also Meredith,
Born in
Africa.
(7)
Some of the research in the Blombos
and Sibudu caves is described in Christopher
Henshilwood, Francesco d’Errico, and Ian Watts,
“Engraved Ochres from the Middle Stone Age
Levels at Blombos Cave, South Africa,” Journal of Human
Evolution 57 (2009): 27–47, as
well as Lucinda Backwell, Francesco d’Errico,
and Lyn Wadley, “Middle Stone Age Bone Tools
from the Howiesons Poort Layers, Sibudu Cave,
South Africa,” Journal
of Archaeological Science 35
(2008): 1566–1580. The location of the African
exodus is taken from the synthesis in Meredith,
Born in
Africa.
(8)
The antiquity of humans in South
America, more specifically, Monte Verde in
present-day Chile, is discussed in David
Meltzer, Donald Grayson, Gerardo Ardila, Alex
Barker, Dena Dincauze, C. Vance Haynes,
Francisco Mena, Lautaro Nunez, and Dennis
Stanford, “On the Pleistocene Antiquity of Monte
Verde, Southern Chile,” American Antiquity 62 (1997):
659–663.
(9)
The cooperative foundation of
language is underscored in, for example, Michael
Tomasello and Esther Herrmann, “Ape and Human
Cognition: What’s the Difference?” Current Directions in
Psychological Science 19 (2010):
3–8, and Michael Tomasello and Amrisha Vaish,
“Origins of Human Cooperation and Morality,”
Annual Review of
Psychology 64 (2013):
231–255.
(10)
For more on how language impacts
thought, see, for example, Caleb Everett,
Linguistic
Relativity: Evidence across Languages and
Cognitive Domains (Berlin: De
Gruyter Mouton, 2013) or Gary Lupyan and
Benjamin Bergen, “How Language Programs the
Mind,” Topics in
Cognitive Science 8 (2016):
408–424.
(11)
For a global survey of world color
terms, see Paul Kay, Brent Berlin, Luisa Maffi,
William Merrifield, and Richard Cook, World Color Survey
(Chicago: University of Chicago Press, 2011). Th
e experimental research conducted among the
Berinmo is reported in Jules Davidoff, Ian
Davies, and Debi Roberson, “Is Color
Categorisation Universal? New Evidence from a
Stone-Age Culture. Colour Categories in a
Stone-Age Tribe,” Nature 398 (1999):
203-204.
(12)
Other terminological choices can be
made here. One could refer to regular quantities
as ‘numbers,’ rather than restricting the usage
of the latter term to words and other symbols
for quantities. If that terminological choice
were adopted, however, the central point would
be unaltered: Our recognition of precise
quantities is largely dependent on number
words.
(13)
Heike Wiese, Numbers, Language, and the
Human Mind (Cambridge: Cambridge
University Press, 2003),
762.
الفصل الثاني: الأعداد منقوشة في ماضينا
(1)
The paintings at Monte Alegre are
discussed in, for example, Anna Roosevelt,
Marconales Lima da Costa, Christiane Machado,
Mostafa Michab, Norbert Mercier, Hélène Valladas,
James Feathers, William Barnett, Maura da Silveira,
Andrew Henderson, Jane Silva, Barry Chernoff, David
Reese, J. Alan Holman, Nicholas Toth, and Kathy
Schick, “Paleoindian Cave Dwellers in the Amazon:
The Peopling of the Americas,” Science 33 (1996):
373–384. For a discussion of the possible
calendrical functions of the particular painting
mentioned here, see Christopher Davis, “Hitching
Post of the Sky: Did Paleoindians Paint an Ancient
Calendar on Stone along the Amazon River?” Proceedings of the Fine
International Conference on Gigapixel Imaging
for Science 1 (2010): 1–18. As Davis
notes, famous nineteenth-century naturalist Alfred
Wallace mentioned and sketched some of these Monte
Alegre paintings in his work.
(2)
The antler was first described in John
Gifford and Steven Koski, “An Incised Antler
Artifact from Little Salt Spring,” Florida Anthropologist
64 (2011): 47–52. The authors of that study note the
possibility that the antler served a calendrical
purpose, though some of the points made here are
based on my own
interpretation.
(3)
Karenleigh Overmann, “Material
Scaffolds in Numbers and Time,” Cambridge Archaeological
Journal 23 (2013): 19–39. For one
comprehensive interpretation of the Taï plaque, see
Alexander Marshack, “The Taï Plaque and Calendrical
Notation in the Upper Paleolithic,” Cambridge Archaeological
Journal 1 (1991):
25–61.
(4)
For one analysis of the Ishango bone,
see Vladimir Pletser and Dirk Huylebrouck, “Th e
Ishango Artefact: The Missing Base 12 Link,”
Forma 14
(1999): 339–346.
(5)
The Lebombo bone is discussed in
Francesco d’Errico, Lucinda Backwell, Paola Villa,
Ilaria Degano, Jeannette Lucejko, Marion Bamford,
Thomas Higham, Maria Colombini, and Peter Beaumont,
“Early Evidence of San Material Culture Represented
by Organic Artifacts from Border Cave, South
Africa,” Proceedings of the
National Academy of Sciences USA 109
(2012): 13214–13219.
(6)
For more on the world’s tally systems,
see Karl Menninger, Number Words
and Number Symbols (Cambridge, MA:
MIT Press, 1969). For a more detailed description of
the Jarawara tally system, see Caleb Everett,
“A Closer Look at a
Supposedly Anumeric Language,” International Journal of American
Linguistics 78 (2012):
575–590.
(7)
For detailed analysis of these
geoglyphs, see Martti Parssinen, Denise Schaan, and
Alceu Ranzi, “Pre-Columbian Geometric Earthworks in
the Upper Purus: A Complex Society in Western
Amazonia,” Antiquity 83 (2009):
1084–1095.
(8)
Karenleigh Overmann, “Finger-Counting
in the Upper Paleolithic,” Rock Art Research 31 (2014):
63–80.
(9)
The Indonesian cave paintings, possibly
the oldest uncovered to date, are discussed in
Maxime Aubert, Adam Brumm, Muhammad Ramli, Thomas
Sutikna, Wahyu Saptomo, Budianto Hakim, Michael
Morwood, G. van den Bergh, Leslie Kinsley, and
Anthony Dosseto, “Pleistocene Cave Art from
Sulawesi, Indonesia,” Nature 514 (2014): 223–227. For an
example of how such cave paintings are dated, see
the discussion of the Fern Cave in Rosemary Goodall,
Bruno David, Peter Kershaw, and Peter Fredericks,
“Prehistoric Hand Stencils at Fern Cave, North
Queensland (Australia): Environmental and
Chronological Implications of Rama Spectroscopy and
FT-IR Imaging Results,” Journal of Archaeological Science 36
(2009): 2617–2624.
(10)
Many books have been written on the
history of writing. My claims here are based in
part on Barry Powell, Writing: Theory and History of the
Technology of Civilization (West
Sussex: Wiley-Blackwell,
2012).
(11)
I am grateful to an anonymous
reviewer for pointing out this
example.
(12)
For more on this Sumerian history,
and the history of other numeral and counting
systems, see Graham Flegg, Numbers through the
Ages (London: Macmillan, 1989)
and Graham Flegg, Numbers: Their History and
Meaning (New York: Schocken
Books, 1983).
(13)
For a cognitively oriented survey
of the world’s numeral systems, see Stephen
Chrisomalis, “A Cognitive Typology for Numerical
Notation,” Cambridge
Archaeological Journal 14 (2004):
37–52.
(14)
The decipherment of Maya writing is
detailed in Michael Coe, Breaking the Maya Code (London:
Th ames & Hudson,
2013).
(15)
Mayan numerals are vigesimally
based, but some calendrical numerals use dots in
the third position to represent 360 instead of
400, that is, they are a combination of base-20
and base-18 patterns. This so-called long-count
system facilitated the specification of dates
with respect to the creation of the universe in
Mayan mythology.
(16)
Th is discussion of numerals only
touches on a few of the ways in which numeral
systems vary, ways that are particularly
relevant for this book. For the most
comprehensive and detailed look at the way
numerals vary, see Stephen Chrisomalis,
Numerical Notation:
A Comparative History (New York:
Cambridge University Press, 2010). Chrisomalis’s
work exhaustively categorizes numeral types
according to a variety of functional
parameters.
(17)
The single knot at the bottom of
the cords, in the ‘ones’ position, represented
different numbers in accordance with how many
loops were needed to make it. In this way, it
was clear that this position represented the
“end” of the numeral. The remaining knots were
simpler and occurred in clusters in the
positions associated with particular exponents.
The account I present here admittedly glosses
over some of the complexity of this semiotic
system, focusing on its decimal nature. For more
on Incan numerals, see, for example, Gary Urton,
“From Middle Horizon Cord-Keeping to the Rise of
Inka Khipus in the Central Andes,” Antiquity 88
(2014): 205–221.
(18)
Flegg, Numbers through the
Ages.
الفصل الثالث: رحلة عددية حول العالم اليوم
(1)
The claim that Jarawara was anumeric
was made in R. M. W. Dixon,
The Jarawara Language of
Southern Amazonia (Oxford: Oxford
University Press, 2004), 559. I describe Jarawara
numbers in Caleb Everett, “A Closer Look at a
Supposedly Anumeric Language,” International Journal of American
Linguistics 78 (2012): 575–590,
583.
(2)
Cardinal number words like ‘one,’
‘two,’ and ‘three’ describe sets of quantities, in
contrast to ordinal words like ‘first,’ ‘second,’
and ‘third.’
(3)
For more formal definitions of bases,
see, for example, Bernard Comrie, “The Search for
the Perfect Numeral System, with Particular
Reference to Southeast Asia,” Linguistik Indonesia 22
(2004): 137–145, or Harald Hammarström, “Rarities in
Numeral Systems,” in Rethinking Universals: How Rarities Affect
Linguistic Theory, ed. Jan Wohlgemuth
and Michael Cysouw (Berlin: De Gruyter Mouton,
2010), 11–59, 15, or Frans Plank, “Senary Summary So
Far,” Linguistic
Typology 3 (2009): 337–345. Such
formal definitions are avoided here as they differ
from one another in minor ways that are not central
to our story.
(4)
The frequency-based reduction of words
is discussed, for instance, in Joan Bybee, The Phonology of Language
Use (Cambridge: Cambridge University
Press, 2001).
(5)
The finger basis of many spoken numbers
is outlined in multiple works, including Alfred
Majewicz, “Le Rôle du Doigt et de la Main et Leurs
Désignations dans la Formation des Systèmes
Particuliers de Numération et de Noms de Nombres
dans Certaines Langues,” in La Main et les Doigts, ed. F. de
Sivers (Leuven, Belgium: Peeters, 1981),
193–212.
(6)
The numbers of languages in particular
families are taken from M. Paul Lewis, Gary Simons,
and Charles Fennig, eds., Ethnologue: Languages of the World,
nineteenth edition (Dallas, TX: SIL International,
2016).
(7)
The word list and discussion of
Indo-European forms is based on Robert Beekes,
Comparative
Indo-European Linguistics: An
Introduction (Amsterdam: John
Benjamins, 1995).
(8)
Andrea Bender and Sieghard Beller,
“‘Fanciful’ or Genuine? Bases and High Numerals in
Polynesian Number Systems,” Journal of the Polynesian Society
115 (2006): 7–46. See as well the discussion of
Austronesian bases in Paul Sidwell, The Austronesian
Languages, revised Edition (Canberra:
Australian National University,
2013).
(9)
This insightful point was made by an
anonymous reviewer.
(10)
Bernard Comrie, “Numeral Bases,” in
The World Atlas of
Language Structures Online, ed.
Matthew Dryer and Martin Haspelmath (Leipzig: Max
Planck Institute for Evolutionary Anthropology,
2013), http://wals.info/chapter/131.
For the most comprehensive survey of the world’s
verbal number systems, see the massive online
database maintained by linguist Eugene Chan:
https://mpi-lingweb.shh.mpg.de/numeral/.
(11)
This point is made in David Stampe,
“Cardinal Number Systems,” in Papers from the Twelft h Regional
Meeting, Chicago Linguistic Society
(Chicago: Chicago Linguistic Society, 1976),
594–609, 596.
(12)
Bernd Heine, The Cognitive Foundations of Grammar
(Oxford: Oxford University Press 1997),
21.
(13)
For more details on the mechanics of
number creation, see James Hurford, Language and Number: Emergence of
a Cognitive System (Oxford:
Blackwell, 1987).
(14)
The “basic numbers” referred to here
are, defined pithily, cardinal terms used to
describe the quantities of sets of
items.
(15)
I am not the first to suggest that
numbers serve as cognitive tools. This point has
been advanced in several works, perhaps most clearly
in Heike Wiese, “The Co-Evolution of Number Concepts
and Counting Words,” Lingua 117 (2007): 758–772, and
Heike Wiese, Numbers,
Language, and the Human Mind
(Cambridge: Cambridge University Press,
2003).
(16)
The Indian merchant counting
strategy is discussed in Georges Ifrah,
The Universal
History of Numbers: From Prehistory to the
Invention of the Computer
(London: Harville Press, 1998). It has also been
suggested that base-60 strategies are due to a
combination of decimal and base-6 systems, in
which case they would still be partially based
on human digits.
(17)
For an analysis of Oksapmin
counting, see Geoffrey Saxe, “Developing Forms
of Arithmetical Thought among the Oksapmin of
Papua New Guinea,” Developmental Psychology 18
(1982): 583–594. Counting among the Yupno is
described in Jurg Wassman and Pierre Dasen,
“Yupno Number System and Counting,” Journal of Cross-Cultural
Psychology 25 (1994):
78–94.
(18)
An overview of base-6 systems is
given in Plank, “Senary Summary So Far.” See
also Mark Donohue, “Complexities with Restricted
Numeral Systems,” Linguistic Typology 12 (2008):
423–429, as well as Nicholas Evans, “Two
pus One
Makes Thirteen: Senary Numerals in the
Morehead-Maro Region,” Linguistic Typology 13 (2009):
321–335.
(19)
See Patience Epps, “Growing a
Numeral System: The Historical Development of
Numerals in an Amazonian Language Family,”
Diachronica 23 (2006): 259–288,
268.
(20)
These points are based in part on
Hammarström, “Rarities in Numeral Systems,”
which surveys rare number bases in the world’s
languages.
(21)
Claims of the limits of numbers in
Australian languages are made in Kenneth Hale,
“Gaps in Grammar and Culture,” in Linguistics and Anthropology:
In Honor of C. F. Voegelin, ed.
M. Dale Kinkade, Kenneth Hale, and Oswald Werner
(Lisse: Peter de Ridder Press, 1975), 295–315,
and R. M. W. Dixon, The
Languages of Australia
(Cambridge: Cambridge University Press, 1980).
The detailed survey of Australian numbers
discussed here is in Claire Bowern and Jason
Zentz, “Diversity in the Numeral Systems of
Australian Languages,” Anthropological Linguistics 54
(2012): 133–160. Despite the relatively
restricted number inventories of Australian
languages, the majority of them also have
grammatical means of expressing concepts like
plural, singular, and even dual, meaning that
their speakers frequently refer to discrete
differences between smaller quantities though
they have limited means of conveying minor
discrepancies between larger quantities. Given
that some Amazonian languages lack the latter
sorts of grammatical means of encoding basic
numerical concepts, and given that the most
restricted number systems are found in Amazonian
languages, it is fair to say that the most
linguistically anumeric groups reside in
Amazonia.
(22)
See Nicholas Evans and Stephen
Levinson, “The Myth of Language Universals:
Language Diversity and Its Importance for
Cognitive Science,” Behavioral and Brain Sciences 32
(2009): 429–448.
(23)
In this chapter we have discussed
global patterns in cardinal numbers, words that
describe the quantities of sets of items. The
focus has been on the representation of words
for positive integers, since other numbers (like
fractions and negative numbers) are less common
in the world’s cultures and are also
comparatively recent innovations. It is worth
mentioning, though, that many generalizations we
have highlighted also apply to fractions, given
that these are based on integers in any given
language. In English, for instance, fractions
such as one tenth, one fifth, and so on, are
inverted units taken from the basic decimal
scale. This is not surprising, since it would be
symbolically cumbersome to switch to, say, a
senary base from a decimal one when speaking
about fractions.
الفصل الرابع: ما بعد مُفردات الأعداد: أنواع أخرى من اللغة العددية
(1)
See Matthew Dryer, “Coding of
Nominal Plurality,” in The World Atlas of Language Structures
Online, ed. Matthew Dryer and
Martin Haspelmath (Leipzig: Max Planck Institute
for Evolutionary Anthropology, 2013),
http://wals.info/chapter/33.
(2)
Stanislas Dehaene, The Number Sense: How the Mind
Creates Mathematics (New York:
Oxford University Press, 2011),
80.
(3)
Some morphological particulars in
Kayardild are glossed over here. For more on the
dual in this language, consult the following
comprehensive grammatical description: Nicholas
Evans, A Grammar of
Kayardild (Berlin: Mouton de
Gruyter, 1995), 184.
(4)
As an anonymous reviewer points
out, some controversial claims of quadral
markers, used in restricted contexts, have been
made for the Austronesian languages Tangga,
Marshallese, and Sursurunga. See the discussion
of these forms in Corbett, Number, 26–29. As
Corbett notes in his comprehensive survey, the
forms are probably best considered paucal
markers. In fact, his impressive survey did not
uncover any cases of quadral marking in the
world’s languages.
(5)
Boumaa Fijian grammatical number is
discussed in R. M. W. Dixon, A Grammar of Boumaa
Fijian (Chicago: University of
Chicago Press, 1988).
(6)
For a book-length discussion of
grammatical number, see Corbett, Number.
(7)
John Lucy, Grammatical Categories and Cognition: A
Case Study of the Linguistic Relativity
Hypothesis (Cambridge: Cambridge
University Press, 1992),
54.
(8)
Caleb Everett, “Language Mediated
Thought in ‘Plural’ Action Perception,” in
Meaning, Form, and
Body, ed. Fey Parrill, Vera
Tobin, and Mark Turner (Stanford, CA: CSLI
2010), 21–40. Note that the pattern described
here is not the same as a verb agreeing with
nominal number. The pattern in question is more
similar to the stampede vs. run example, in
which a verb has inherent plural
connotations.
(9)
Dehaene, The Number
Sense.
(10)
For evidence of the commonality of
1–3, see Frank Benford, “The Law of Anomalous
Numbers,” Proceedings of
the American Philosophical
Society 78 (1938): 551–572. For a
discussion of the commonality of smaller
quantities and of multiples of 10, see Dehaene,
The Number
Sense,
99–101.
(11)
This example of Roman numerals has
been noted elsewhere, for instance, in Dehaene,
The Number
Sense.
(12)
The range of sounds in languages is
taken from Peter Ladefoged and Ian Maddieson,
The Sounds of the
World’s Languages (Hoboken, NJ:
Wiley Blackwell, 1996). For one study on the
potential environmental adaptations of
languages, see Caleb Everett, Damián Blasi, and
Seán Roberts, “Climate, Vocal Cords, and Tonal
Languages: Connecting the Physiological and
Geographic Dots,” Proceedings of the National Academy of
Sciences USA 112 (2015):
1322–1327.
الجزء الثاني: عوالم بلا أعداد
الفصل الخامس: شعوب لا عددية مُعاصرة
(1)
The Pirahã have been discussed
extensively elsewhere, most notably in my father’s
book: Daniel Everett, Don’t
Sleep, There Are Snakes: Life and
Language in the
Amazonian Jungle (New York: Random
House, 2008).
(2)
John Hemming, Tree of Rivers: The Story of the
Amazon (London: Thames and Hudson,
2008), 181.
(3)
In fact, he became a very well-known
scholar after encountering the Pirahã and has
published numerous works on their language as well
as other topics. These works have led to extensive
discussion in academic circles, and in the media, on
the nature of language. Most famously, perhaps, his
research on the language suggests that the Pirahã
language lacks recursion, a syntactic feature
assumed by some linguists to occur in all
languages.
(4)
These results on the imprecision of
number-like words in the language are presented in
Michael Frank, Daniel Everett, Evelina Fedorenko,
and Edward Gibson, “Number as a Cognitive
Technology: Evidence from Pirahã Language and
Cognition,” Cognition 108 (2008): 819–824. My
discussion combines the results of the “increasing
quantity elicitation” and “decreasing quantity
elicitation” tasks in that study. The observation
that all number-like words in the language are
imprecise was offered earlier, in Daniel Everett,
“Cultural Constraints on Grammar and Cognition in
Pirahã: Another Look at the Design Features of Human
Language,” Current
Anthropology 46 (2005):
621–646.
(5)
Pierre Pica, Cathy Lemer, Veronique
Izard, and Stanislas Dehaene, “Exact and Approximate
Arithmetic in an Amazonian Indigene Group,”
Science 306
(2004): 499–503.
(6)
Peter Gordon, “Numerical Cognition
without Words: Evidence from Amazonia,”
Science
36 (2004): 496–499.
(7)
In other words, the correlation had
what psychologists call a standard coefficient of
variation. The coeffi cient of
variation refers to the ratio one arrives at by
taking the standard deviation of responses and
dividing it by the correct responses, for each
target quantity. Gordon found that the
coefficient of variation hovered around 0.15 for
all quantities greater than three. We observed
the same pattern in follow-up work among the
Pirahã.
(8)
See Caleb Everett and Keren Madora,
“Quantity Recognition among Speakers of an
Anumeric Language,” Cognitive Science 36 (2012):
130–141.
(9)
The results obtained at Xaagiopai
do suggest that, when the Pirahã have had some
practice with number words in their own
language, they also begin to show signs of
recognizing larger quantities more precisely.
After all, their performance on the basic line
matching task did seem to improve in that
village after some number-word
familiarization.
(10)
Interestingly, some languages in
South Australia have “birth-order names,” which
indicate someone’s relative age when contrasted
to their siblings. As an anonymous reviewer
points out, this is true in the Kaurna language,
for example.
(11)
These Munduruku findings are
presented in Pica et al., “Exact and Approximate
Arithmetic in an Amazonian Indigene
Group.”
(12)
Pica et al., “Exact and Approximate
Arithmetic in an Amazonian Indigene Group,”
502.
(13)
Franc Marušič, Rok Žaucer, Vesna
Plesničar, Tina Razboršek, Jessica Sullivan, and
David Barner, “Does Grammatical Structure Speed
Number Word Learning? Evidence from Learners of
Dual and Non-Dual Dialects of Slovenian,”
PLoS ONE
11 (2016): e0159208.
doi:10.1371/journal.pone.0159208.
(14)
Stanislas Dehaene, The Number Sense: How the Mind
Creates Mathematics (New York:
Oxford University Press, 2011), 264.
(15)
Koleen McCrink, Elizabeth Spelke,
Stanislas Dehaene, and Pierre Pica,
“Non-Developmental Halving in an Amazonian
Indigene Group,” Developmental Science 16 (2012):
451–462.
(16)
Maria de Hevia and Elizabeth
Spelke, “Number-Space Mapping in Human I
nfants,” Psychological
Science 21 (2010): 653–660.
(17)
The study of the mental number line
evident among the
Munduruku is Stanislas
Dehaene, Veronique Izard, Elizabeth Spelke, and
Pierre Pica, “Log or Linear? Distinct Intuitions
of the Number Scale in Western and Amazonian
Indigene Cultures,” Science 320 (2008): 1217–1220.
(18)
Rafael Núñez, Kensy Cooperrider,
and Jurg Wassman, “Number Concepts without
Number Lines in an Indigenous Group of Papua New
Guinea,” PLoS
ONE 7 (2012): 1–8.
(19)
Elizabet Spaepen, Marie Coppola,
Elizabeth Spelke, Susan
Carey, and Susan
Goldin-Meadow, “Number without a Language
Model,” Proceedings of
the National Academy of Sciences
USA 108 (2011): 3163–3168,
3167.
(20)
Only now are there signs that
pressures from the outside will eventually yield
the systematic adoption of numbers into these
cultures. For instance, many governmental
resources have recently been dedicated to
familiarizing the Pirahã at Xaagiopai with
Portuguese, including Portuguese number words.
الفصل السادس: الكميات في عقول الأطفال الصغار
(1)
We do not know when exactly these
number senses become accessible to us, though as we
shall see, the approximate number sense is
accessible at birth. My reference to number ‘senses’
owes itself to Stanislas Dehaene’s fantastic book,
The Number Sense: How
the Mind Creates Mathematics (New
York: Oxford University Press, 2011). As first noted
in Chapter 4, the exact number sense is actually
enabled by a more general capacity for tracking
discrete objects. The quantitative function of this
capacity is epiphenomenal. For mnemonic ease I refer
to this quantitative function as the exact number
sense, as it is what enables the relatively precise
differentiation of smaller sets of items. For more
on the general object-tracking or “parallel
individuation” capacity that enables the
discrimination of small quantities, see, for
example, Elizabeth Brannon and Joonkoo Park,
“Phylogeny and Ontogeny of Mathematical and
Numerical Understanding,” in The Oxford Handbook of Numerical
Cognition, ed. Roy Cohen Kadosh and
Ann Dowker (Oxford: Oxford University Press, 2015),
203–213.
(2)
One case for an innate language
capacity is elegantly presented in Steven
Pinker, The Language
Instinct: The New Science of Language and
Mind (London: Penguin Books,
1994). For more recent alternative perspectives,
the reader may wish to consult accessible texts
such as Vyv Evans, The
Language Myth: Why Language Is Not an
Instinct (Cambridge: Cambridge
University Press, 2014) or Daniel Everett,
Language: The
Cultural Tool (New York: Random
House, 2012).
(3)
Karen Wynn, “Addition and
Subtraction by Human Infants,” Nature 358 (1992):
749–750.
(4)
Furthermore, the study addressed some
of the criticisms leveled at Wynn, “Addition and
Subtraction by Human Infants,” as well as other
studies that did not control for non-numerical
confounds like amount, shape, and confi guration
of stimuli. See Fei Xu and Elizabeth Spelke,
“Large Number Discrimination in 6-Month-Old
Infants,” Cognition 74 (2000):
B1-B11.
(5)
I say “most infants” here, because
for four of the sixteen infants who participated
in the study, no staring differences were
observed when they encountered novel amounts of
dots.
(6)
Xu and Spelke, “Large Number
Discrimination in 6-Month-Old Infants,”
B10.
(7)
This is an understandable issue
with psychological research more generally,
which is typically focused on peoples in
Western, educated, and industrialized societies,
since such peoples are easily accessible to most
psychologists. See the discussion in Joseph
Henrich, Steven Heine, and Ara Norenzayan, “The
Weirdest People in the World?” Behavioral and Brain
Sciences 33 (2010): 61–83.
(8)
The study described here is
Veronique Izard, Coralie Sann, Elizabeth Spelke,
and Arlette Streri, “Newborn Infants Perceive
Abstract Numbers,” Proceedings of the National Academy of
Sciences USA 106 (2009):
10382–10385.
(9)
Such evidence does not suggest,
however, that the human brain is uniquely hardwired
for mathematical thought. As we will see in
Chapter 7, other species also have an abstract
number sense for differentiating quantities when
the ratio between them is sufficiently
large.
(10)
Jacques Mehler and Thomas Bever,
“Cognitive Capacity of Very Young Children,”
Science
3797 (1967): 141-142. See also the enlightening
discussion on this topic in Dehaene, The Number Sense: How the Mind
Creates Mathematics, particularly
as it relates to the work of Piaget. I should
mention, however, that an insightful reviewer
notes that there have been issues replicating
the results of Mehler and Bever with very young
children.
(11)
Kirsten Condry and Elizabeth
Spelke, “The Development of Language and
Abstract Concepts: The Case of Natural Number,”
Journal of
Experimental Psychology: General
137 (2008): 22–38.
(12)
For a different perspective, see
Rochel Gelman and C. Randy Gallistel, Young Children’s Understanding
of Numbers (Cambridge, MA:
Harvard University Press, 1978), or Rochel
Gelman and Brian Butterworth, “Number and
Language: How Are They Related?” Trends in Cognitive
Sciences 9 (2005): 6–10. Note
that these works predate some of the research
discussed here.
(13)
A more detailed discussion of the
successor principle is presented in, for
example, Barbara Sarnecka and Susan Carey, “How
Counting Represents Number: What Children Must
Learn and When They Learn It,” Cognition 108
(2008): 662–674.
(14)
For more on the acquisition of
these concepts by children in numerate cultures,
I refer the reader to Susan Carey, The Origin of
Concepts (Oxford: Oxford
University Press, 2009), and Susan Carey, “Where
Our Number Concepts Come From,” Journal of
Philosophy 106 (2009):
220–254.
(15)
See Elizabeth Gunderson, Elizabet
Spaepen, Dominic Gibson, Susan Goldin-Meadow,
and Susan Levine, “Gesture as a Window onto
Children’s Number Knowledge,” Cognition 144
(2015): 14–28, 22.
(16)
See Barbara Sarnecka, Megan
Goldman, and Emily Slusser, “How Counting Leads
to Children’s First Representations of Exact,
Large Numbers,” in The
Oxford Handbook of Numerical
Cognition, ed. Roy Cohen Kadosh
and Ann Dowker (Oxford: Oxford University Press,
2015), 291–309. For more on the acquisition of
one-to-one correspondence, see also Barbara
Sarnecka and Charles Wright, “The Idea of an
Exact Number: Children’s Understanding of
Cardinality and Equinumerosity,” Cognitive Science
37 (2013): 1493–1506.
(17)
See Carey, The Origin of Concepts. Carey’s
account suggests that the innate exact
differentiation of small quantities is the chief
facilitator of the acquisition of other
numerical concepts. In other words, the
approximate number sense plays a less
substantive role in the initial structuring of
numbers, when contrasted to some other accounts.
Some empirical support for her account is
offered, for instance, in Mathiew Le Corre and
Susan Carey, “One, Two, Three, Four, Nothing
More: An Investigation of the Conceptual Sources
of the Verbal Counting Principles,” Cognition 105
(2007): 395–438. Debate remains among
specialists as to how our innate number senses
are fused. But it is generally agreed that both
contribute to the eventual acquisition of
numerical and arithmetical concepts.
(18)
The phrase “concepting labels” is
taken from Nick Enfield, “Linguistic Categories
and Their Utilities: The Case of Lao Landscape
Terms,” Language
Sciences 30 (2008): 227–255, 253.
For more on the way that number words serve as
placeholders for concepts in the minds of kids,
see Sarnecka, Goldman, and Slusser, “How
Counting Leads to Children’s First
Representations of Exact, Large Numbers.”
(19)
While truly representative
cross-cultural studies on the development of
numerical thought are largely missing in the
literature, some recent work with a
farming-foraging culture in the Bolivian
rainforest, the Tsimane’, explores these issues.
The Tsimane’ take about two to three times as
long to learn to count, when contrasted with
children in industrialized societies. See Steve
Piantadosi, Julian Jara-Ettinger, and Edward
Gibson, “Children’s Learning of Number Words in
an Indigenous Farming-Foraging Group,” Developmental
Science 17 (2014): 553–563. A
very recent study of this group has found that
their understanding of exact quantity
correspondence correlates with knowledge of
numbers and counting, as predicted by the
account presented here. Interestingly, however,
that same study suggests that there is at least
one Tsimane’ child “who cannot count but
nevertheless understands the logic of exact
equality.” This is unexpected but not startling
either. Aft er all, we know that some humans
(like number inventors) come to recognize exact
equality without fi rst counting. Of course,
these Tsimane’ kids still have exposure to
counting and numerical semiotic practices, as
they are embedded in a numerate culture. It is
clear from all the relevant work, including that
among the Tsimane’, that learning to count
greatly facilitates the subsequent recognition
of precise quantities. See Julian Jara-Ettinger,
Steve Piantadosi, Elizabeth S. Spelke, Roger
Levy, and Edward Gibson, “Mastery of the Logic
of Natural Numbers is not the Result of Mastery
of Counting: Evidence form Late Counters,”
Developmental
Science 19 (2016): 1–11.
doi:10.1111/desc12459, 8.
الفصل السابع: الكميَّات في عقول الحيوانات
(1)
For more on this experiment, of which I
have provided only a basic summary, see Daniel
Hanus, Natacha Mendes, Claudio Tennie, and Josep
Call, “Comparing the Performances of Apes (Gorilla gorilla, Pan troglodytes,
Pongo pygmaeus) and Human Children
(Homo
sapiens) in the Floating Peanut
Task,” PLoS ONE 6
(2011): e19555.
(2)
For evidence on the extent to which the
collaboration between animals and humans impacted
our species, see Pat Shipman, “The Animal Connection
and Human Evolution,” Current Anthropology 54 (2010):
519–538.
(3)
For more on Clever Hans, see Oscar
Pfungst, Clever Hans: (The
Horse of Mr. von Osten) A Contribution to Animal
and Human Psychology (New York: Holt
and Company, 1911).
(4)
See Charles Krebs, Rudy Boonstra,
Stan Boutin, and A. R. E. Sinclair, “What Drives
the 10-Year Cycle of Snowshoe Hares?” Bioscience 51
(2001): 25–35.
(5)
The emergence of prime numbers in
such cycles is described in Paulo Campos,
Viviane de Oliveira, Ronaldo Giro, and Douglas
Galvão, “Emergence of Prime Numbers as the
Result of Evolutionary Strategy,” Physical Review
Letters 93 (2004): 098107.
(6)
Nevertheless, it must be
acknowledged that some invertebrate species
exhibit behaviors consistent with rudimentary
quantity approximation. See the survey in
Christian Agrillo, “Numerical and Arithmetic
Abilities in Non-Primate Species,” in Oxford Handbook of Numerical
Cognition, ed. Ann Dowker
(Oxford: Oxford University Press, 2015),
214–236.
(7)
The numerical cognition of
salamanders is described in Claudia Uller,
Robert Jaeger, Gena Guidry, and Carolyn Martin,
“Salamanders (Plethodon
cinereus) Go for More: Rudiments
of Number in an Amphibian,” Animal Cognition 6
(2003): 105–112, and also in Paul Krusche,
Claudia Uller, and Ursula Dicke, “Quantity
Discrimination in Salamanders,” Journal of Experimental
Biology 213 (2010): 1822–1828.
Results obtained with fish are described in
Christian Agrillo, Laura Piffer, Angelo Bisazza,
and Brian Butterworth, “Evidence for Two
Numerical Systems That Are Similar in Humans and
Guppies,” PLoS
ONE 7 (2012): e31923.
(8)
The seminal study of rats is that
of John Platt and David Johnson, “Localization
of Position within a Homogeneous Behavior Chain:
Effects of Error Contingencies,” Learning and
Motivation 2 (1971): 386–414.
(9)
Regarding lionesses, see Karen
McComb, Craig Packer, and Anne Pusey, “Roaring
and Numerical Assessment in the Contests between
Groups of Female Lions, Panther leo,” Animal Behaviour 47
(1994): 379–387. For findings on pigeons, see
Jacky Emmerton, “Birds’ Judgments of Number and
Quantity,” in Avian
Visual Cognition, ed. Robert Cook
(Boston: Comparative Cognition Press, 2001).
(10)
Agrillo, “Numerical and Arithmetic
Abilities in Non-Primate Species,” 217.
(11)
Results vis-à-vis dogs are offered
in Rebecca West and Robert Young, “Do Domestic
Dogs Show Any Evidence of Being Able to Count?”
Animal
Cognition 5 (2002): 183–186. For
findings with robins, see Simon Hunt, Jason Low,
and K. C. Burns, “Adaptive Numerical Competency
in a Food-Hoarding Songbird,” Proceedings of the Royal
Society of London: Biological
Sciences 267 (2008):
2373–2379.
(12)
Agrillo et al., “Evidence for Two
Numerical Systems That Are Similar in Humans and
Guppies.”
(13)
The similarity of the human and
chimp genomes is described by The Chimpanzee
Sequencing and Analysis Consortium, “Initial
Sequence of the Chimpanzee Genome and Comparison
with the Human Genome,” Nature 437 (2005): 69–87. The
value of genomic correspondence varies depending
on the methods used, but is generally found to
be greater than 95 percent. See also Roy
Britten, “Divergence between Samples of
Chimpanzee and Human DNA Sequences is 5%
Counting Indels,” Proceedings of the National Academy of
Sciences USA 99 (2002):
13633–13635. For an exploration of the human
genetic similarity to other species, visit
http://ngm.nationalgeographic.com/2013/07/125-explore/shared-genes.
(14)
Mihaela Pertea and Steven Salzberg,
“Between a Chicken and a Grape: Estimating the
Number of Human Genes,” Genome Biology 11 (2010): 206.
(15)
See Marc Hauser, Susan Carey, and
Lilan Hauser,
“Spontaneous
Number Representation in Semi-Free Ranging
Rhesus Monkeys,” Proceedings of the
Royal Society of London: Biological
Science 267 (2000): 829–833. Some
of Hauser’s work has been called into question
due to an inquiry conducted at Harvard, which
found evidence that some of his results had been
tampered with. The results in this particular
study are not involved in that inquiry.
(16)
The results on this ascending task
are described in Elizabeth Brannon and Herbert
Terrace, “Ordering of the Numerosities 1–9 by
Monkeys,” Science 282 (1998):
746–749.
(17)
The chocolate experiment is
described in Duane Rumbaugh,
Sue
Savage-Rumbaugh, and Mark Hegel, “Summation in
the Chimpanzee (Pan
troglodytes),” Journal of Experimental
Psychology: Animal Behaviors
Processes
13 (1987): 107–115.
(18)
Support for these claims is
presented in Brannon and Terrace,
“Ordering of the
Numerosities 1–9 by Monkeys.” With respect to
baboons and squirrel monkeys, see Brian Smith,
Alexander Piel, and Douglas Candland, “Numerity
of a Socially Housed Hamadryas Baboon (Papio hamadryas)
and a Socially Housed Squirrel Monkey (Saimiri sciureus),” Journal
of Comparative Psychology 117
(2003): 217–225. For more on squirrel monkeys,
see Anneke Olthof, Caron Iden, and William
Roberts, “Judgements of Ordinality and Summation
of Number Symbols by Squirrel Monkeys (Saimiri sciureus),” Journal
of Experimental Psychology: Animal Behaviors
Processes 23 (1997): 325–339.
Monkeys are capable of selecting the larger
quantity of food items via approximation or via
more exact methods that depend on training with
numbers. Yet their quantity-discrimination
skills are not restricted to the realm of
consumables. Studies have also shown that rhesus
monkeys can accurately choose the larger of two
digital arrays of items presented via computer
screen, even after non-numeric properties, such
as surface area of the presented stimuli, are
controlled. See Michael Beran, Bonnie Perdue,
and Theodore Evans, “Monkey Mathematical
Abilities,” in Oxford
Handbook of Numerical Cognition,
ed. Ann Dowker (Oxford: Oxford University Press,
2015), 237–259.
(19)
The cross-species evidence for an
exact number sense, enabled by what is often
referred to as the parallel individuation
system, is weaker and, to some researchers,
marginal at best. See discussion in Beran,
Perdue, and Evans, “Monkey Mathematical
Abilities.” Researchers have not fully fleshed
out the range of similarity between our innate
number senses and those evident in other
species, such as our primate
relatives.
(20)
Elizabeth Brannon and Joonkoo Park,
“Phylogeny and Ontogeny of Mathematical and
Numerical Understanding,” in Oxford Handbook of Numerical
Cognition, ed. Ann Dowker
(Oxford: Oxford University Press, 2015),
209.
(21)
Irene Pepperberg, “Further Evidence
for Addition and Numerical Competence by a Grey
Parrot (Psittacus
erithacus),” Animal Cognition 15
(2012): 711–717. For results with Sheba, see
Sarah Boysen and Gary Berntson, “Numerical
Competence in a Chimpanzee (Pan troglodytes),” Journal of
Comparative Psychology 103
(1989): 23–31.
(22)
Pepperberg, “Further Evidence for
Addition and Numerical Competence by a Grey
Parrot (Psittacus
erithacus),” 711.
الجزء الثالث: الأعداد وتشكيل حياتنا
الفصل الثامن: اختراع الأعداد والحساب
(1)
To read more about how patterns in
language impact thought, see Caleb Everett,
Linguistic Relativity:
Evidence across Languages and Cognitive
Domains (Berlin: De Gruyter Mouton,
2013).
(2)
James Hurford, Language and Number: Emergence
of a Cognitive System (Oxford:
Blackwell, 1987), 13. The perspective I present
here is influenced by the more recent work of
Heike Wiese, “The Co-Evolution of Number
Concepts and Counting Words,” Lingua 117 (2007):
758–772. She observes on page 762 that “the dual
status of counting words crucially means that
they are numbers (as well as words), rather than
number names, that is, they do not refer to
extra-linguistic ‘numbers’, but instead are used
as numbers right away.” Wiese also notes that
the traditional “numbers-as-names” approach
overlooks ordinal (‘first,’ ‘second,’ etc.) and
nominal (e.g., “the #9 bus”) number
words.
(3)
Karenleigh Overmann, “Numerosity
Structures the Expression of Quantity in Lexical
Numbers and Grammatical Number,” Current
Anthropology 56 (2015): 638–653,
639. For a reply to this article, see Caleb
Everett, “Lexical and Grammatical Number Are
Cognitive and Historically Dissociable,”
Current
Anthropology 57 (2016): 351.
(4)
Stanislas Dehaene, The Number Sense: How the Mind
Creates Mathematics (New York:
Oxford University Press, 2011), 80.
(5)
See Kevin Zhou and Claire Bowern,
“Quantifying Uncertainty in the Phylogenetics of
Australian Number Systems,” Proceedings of the Royal
Society B: Biological Sciences
282 (2015): 2015–1278. These findings are
consistent with the related discussion of
Australian numbers in Chapter 3, which was based
on a separate study-one also co-authored by
Bowern.
(6)
The physical bases of number words
has been observed in many sources, for instance,
in Bernd Heine, Cognitive Foundations of Grammar
(Oxford: Oxford University Press,
1997).
(7)
Apart from any particular
contestable details of this account, little
doubt remains that number words are verbal
tools, not merely labels for concepts that all
people are innately predisposed to recognize.
See also Wiese, “The Co-Evolution of Number
Concepts and Counting Words,” 769, where she
notes, for example, that “counting words are
verbal instances of numerical tools, that is,
verbal tools we use in number assignments.”
(8)
There are many works on embodied
cognition. For one extensive survey of this
topic, consult Lawrence Shapiro (ed.), The Routledge Handbook of
Embodied Cognition (New York:
Routledge, 2014). In contrast to the account
presented here, some archaeologists have focused
on how body-external features have impacted the
innovation of numbers. See, for example,
Karenleigh Overmann, “Material Scaff olds in
Numbers and Time,” Cambridge Archaeological Journal
23 (2013): 19–39. They suggest an alternate
account, according to which materials like
beads, tokens, and tally marks served as
material placeholders for concepts that were
then instantiated linguistically. No doubt such
artifacts, like other material factors, placed
additional pressures on humans to invent and
refine numbers. (See Chapter 10.) But the
perspective espoused here is that the anatomical
pathways to numbers are more basic
ontogenetically and historically when contrasted
to any other (no doubt extant) external numeric
placeholders. Fingers are, after all, more
experientially primal than such body-external
material stimuli. In addition, there is a clear
tie between numeric language and the body (see
Chapter 3), which suggests the primacy of the
body in inventing numbers, not just labeling
them after material placeholders for numbers are
invented. The claim here is not, however, that
material technologies and symbols do not also
play a role in fostering numerical thought, and
the research of such archaeologists is crucial
to elucidating the extent of that role. As
humans engaged with numbers materially, we no
doubt faced greater pressures to extend our
number systems in new ways. But, even
considering such pressures, our fingers are what
enabled the very invention of numbers, at least
in most cases.
(9)
Rafael Núñez and Tyler Marghetis,
“Cognitive Linguistics and the Concept(s) of
Number,” in The Oxford
Handbook of Numerical Cognition,
ed. Roy Cohen Kadosh and Ann Dowker (Oxford:
Oxford University Press, 2015), 377–401,
377.
(10)
For a detailed consideration of the
role of meta phors in the creation of math, see
George Lakoff and Rafael Núñez, Where Mathe matics Comes From:
How the Embodied Mind Brings Mathematics
into Being (New York: Basic
Books, 2001). For a more recent consideration,
see Núñez and Marghetis, “Cognitive Linguistics
and the Concept(s) of Number.”
(11)
. Núñez and Marghetis, “Cognitive
Linguistics and the Concept(s) of Number,” 402.
(12)
Núñez and Marghetis, “Cognitive
Linguistics and the Concept(s) of Number,” 402.
(13)
Of course, kids are frequently
counting actual objects when they learn and use
math. Yet the larger point is that in all
contexts, including abstract ones, we use a
physical grounding to talk about how we mentally
manipulate the quantities represented through
numbers. Such meta phorical bases of numerical
language are common throughout the world. In
Chapter 5 it was noted, though, that number
lines are not used in all cultures to make sense
of quantities.
(14)
The value of gestures in exploring
human cognition is evident, for example, in
Susan Goldin-Meadow, The
Resilience of Language: What Gesture
Creation in Deaf Children Can Tell Us about
How All Children Learn Language
(New York: Psychology Press, 2003). The findings
on mathematical gestures discussed here are also
taken from Núñez and Marghetis, “Cognitive
Linguistics and the Concept(s) of
Number.”
(15)
These points on brain imaging are
adapted from Stanislas Dehaene, Elizabeth
Spelke, Ritta Stanescu, Philippe Pinel, and
Susanna Tsivkin, “Sources of Mathematical
Thinking: Behavioral and Brain-Imaging
Evidence,” Science 284 (1999): 970–974. The
spatial interference example is adapted from
Dehaene, The Number
Sense: How the Mind Creates
Mathematics,
243.
(16)
This SNARC effect was first
described in Stanislas Dehaene, Serge Bossini,
and Pascal Giraux, “The Mental Representation of
Parity and Number Magnitude,” Journal of Experimental
Psychology: General 122 (1993):
371–396.
(17)
See Heike Wiese, Numbers, Language, and the
Human Mind (Cambridge: Cambridge
University Press, 2003), and Wiese, “The
Co-Evolution of Number Concepts and Counting
Words,” for a detailed account of how syntax may
impact numerical thought. According to Wiese,
this sort of linguistically based thinking
enables us to use not just cardinal numbers,
which refer to the values of particular sets of
items, but also ordinal and nominal numbers.
(See note 2.) Such valuable insights should not
be overextended either. The range of diversity
in the world’s languages should give us pause
before concluding that syntactic influences play
a major role in the expansion of numerical
thought in all cultures. Considering the extent
to which some languages allow so-called free
word order and do not have rigid syntactic
constraints like English, such caution is
prudent. These include many languages with rich
case systems that convey who the subject and
object are irrespective of their position in a
clause (Latin, for instance). The speakers of
some languages with freer syntax still acquire
numbers. This does not imply that syntax does
not play a role in facilitating our own
acquisition of such concepts. However, any
influence of grammar on the way we learn numbers
likely varies substantially across cultures.
(18)
For more on brain-to-body size
ratios, see Lori Marino, “A Comparison of
Encephalization between Ondontocete Cetaceans
and Anthropoid Primates,” Brain, Behavior and
Evolution 51 (1998) 230–238. For
further details of the human cortex, see Suzana
Herculano-Houzel, “The Human Brain in Numbers: A
Linearly Scaled-Up Primate Brain,” Frontiers in Human
Neuroscience 3 (2009):
doi:10.3389/neuro.09.031.2009. The neuron count
used here is taken from Dorte Pelvig, Henning
Pakkenberg, Anette Stark, and Bente Pakkenberg,
“Neocortical Glial Cell Numbers in Human
Brains,” Neurobiology of
Aging 29 (2008):
1754–1762.
(19)
IPS activation in monkeys is
described in Andreas Nieder and Earl Miller, “A
Parieto-Frontal Network for Visual Numerical
Information in the Monkey,” Proceedings of the National
Academy of Sciences USA 19
(2004): 7457–7462. The interaction of cortical
regions and particular quantities has been
discussed in various works, including Dehaene,
The Number Sense:
How the Mind Creates Mathe
matics,
248–251.
(20)
Relevant locations in the IPS are
presented in Stanislas Dehaene, Manuela Piazza,
Philippe Pinel, and Laurent Cohen, “Three
Parietal Circuits for Number Processing,”
Cognitive
Neuropsychology 20 (2003):
487–506. Degree of activation is discussed in
Philippe Pinel, Stanislas Dehaene, D. Rivière,
and Denis LeBihan, “Modulation of Parietal
Activation by Semantic Distance in a Number
Comparison Task,” Neuroimage 14 (2001):
1013–1026.
(21)
See Dehaene, The Number Sense: How the Mind
Creates Mathe matics, 241, for
imaging evidence of the verbal expansion of
quantitative reasoning. Given that the hIPS is
clearly associated with numerical cognition,
some researchers have posited a brain
“module”
dedicated to numerical thought. See Brian
Butterworth, The
Mathematical Brain (London:
Macmillan, 1999). It is important to recall that
the cortex is highly plastic and that, although
certain parts of the brain may be associated
with certain functions, these regions may vary
across individuals.
الفصل التاسع: الأعداد والثقافة: نمَط الإعاشة والرمزية
(1)
Khufu was about 8 meters taller before
its outer shell eroded. Using the original height
(139 + 8), we have 147 × 2 × π = 924, while the
perimeter is 230 × 4 = 920.
(2)
The most widely cited survey of
color terms is Brent Berlin and Paul Kay,
Basic Color Terms:
Their Universality and Evolution
(Berkeley: University of California Press,
1969). Fascinating data on the cross-cultural
variability of olfactory categorizations are
presented in Asifa Majid and Niclas Burenhult,
“Odors are Expressable in Language, as Long as
You Speak the Right Language,” Cognition 130
(2014): 266–270.
(3)
The correlation between numbers and
subsistence strategy is presented in the global
survey in Patience Epps, Claire Bowern, Cynthia
Hansen, Jane Hill, and Jason Zentz, “On Numeral
Complexity in Hunter-Gatherer Languages,”
Linguistic
Typology 16 (2012): 41–109. The
findings on Bardi are taken from the same work,
p. 50.
(4)
As we saw in Chapter 8, however,
some Australian languages do have a number word
for 5, which leads to the relatively rapid
innovation of larger
numbers.
(5)
For more on the isolation of some
Amazonian groups, see Dylan Kesler and Robert
Walker, “Geographic Distribution of Isolated
Indigenous Societies in Amazonia and the
Efficacy of Indigenous Territories,” PLoS ONE 10 (2015):
e0125113.
(6)
Although we should not denigrate
particular linguistic and cultural traditions,
we can avoid such prejudices while
simultaneously acknowledging that numerical
technologies enable certain types of reasoning
that, in turn, yield new kinds of innovations.
These innovations, it should be admitted,
ultimately include such benefits as medicinal
technologies that yield longer life spans. So
even though numbers may not lead to impartially
considered “better” or “more advanced” lives,
they were indubitably crucial to the transition
to longer life spans. Of course numbers were
also crucial to less pleasant developments, such
as mechanized warfare.
(7)
See, for instance, Andrea Bender
and Sieghard Beller, “Mangarevan Invention of
Binary Steps for Easier Calculation,” Proceedings of the National
Academy of Sciences USA 111
(2014): 1322–1327, as well as Andrea Bender and
Sieghard Beller, “Numeral Classifiers and
Counting Systems in Polynesian and Micronesian
Languages: Common Roots and Cultural
Adaptations,” Oceanic
Linguistics 25 (2006): 380–403.
See also Sieghard Beller and Andrea Bender, “The
Limits of Counting: Numerical Cognition between
Evolution and Culture,” Science 319 (2008):
213–215.
(8)
For birth-order names in South
Australian languages, see Rob Amery, Vincent
Buckskin, and Vincent “Jack” Kanya, “A
Comparison of Traditional Kaurna Kinship
Patterns with Those Used in Contemporary Nunga
English,” Australian
Aboriginal Studies 1 (2012):
49–62.
(9)
Bender and Beller, “Mangarevan
Invention of Binary Steps for Easier
Calculation,” 1324.
(10)
For more on the potential
advantages of such technologies, consult, for
example, Michael Frank, “Cross-Cultural
Differences in Representations and Routines for
Exact Number,” Language
Documentation and Conservation 5
(2012): 219–238. See also the survey of
technologies like abaci in Karl Menninger,
Number Words and
Number Symbols (Cambridge, MA:
MIT Press, 1969).
(11)
The recent rediscovery of the
eastern hemi sphere’s oldest zero, in Cambodia,
is described in Amir Aczel, Finding Zero: A
Mathematician’s Odyssey to Uncover the
Origins of Numbers (New York:
Palgrave Macmillan, 2015). Given the heavy
influence of Indian culture on the Khmer, it is
assumed that zero was transferred from India to
Cambodia. Still, the oldest defi nitive instance
of zero in the Old World is that found near
Angkor, first discovered in the 1930s and
rediscovered in 2015 by Aczel-who scoured many
stone stelae to find it.
(12)
For rich surveys of the world’s
written numeral systems, see Stephen
Chrisomalis, Numerical
Notation: A Comparative History
(New York: Cambridge University Press, 2010), as
well as Stephen Chrisomalis, “A Cognitive
Typology for Numerical Notation,” Cambridge Archaeological
Journal 14 (2004):
37–52.
(13)
There is some argument as to
whether Egyptian hieroglyphs were innovated in
dependently of an awareness of writing in
Sumeria. They appear on the scene not long after
the development of Mesopotamian writing, by most
accounts. Given that Sumeria and Egypt are
relatively proximate geograph i cally, it is
likely that Egyptians developed hieroglyphs only
after they became knowledgeable of the existence
of writing.
(14)
For a look at early cuneiform, see
Eleanor Robson, Mathematics in Ancient Iraq: A Social
History (Prince ton, NJ: Prince
ton University Press, 2008). For a discussion of
numbers in early written forms, see Stephen
Chrisomalis, “The Origins and Co-Evolution of
Literacy and Numeracy,” in The Cambridge Handbook of
Literacy, ed. David Olson and
Nancy Torrance (New York: Cambridge University
Press, 2009), 59–74. Chrisomalis describes the
copresence of numerals and ancient writing
systems, though he notes that this copresence
may be coincidental.
(15)
However, I should be clear that tally
systems do not necessarily develop into writing
systems or written numerals. The Jarawara tally
system, pictured in Figure 2.2, did not
eventually yield a native Jarawara system of
writing. The same could be said of some tally
systems that have existed in Africa and
elsewhere for thousands of years. But even
though the existence of a tally system may not
be a sufficient condition for the invention of
writing, it may increase the likelihood of a
writing system being
innovated.
الفصل العاشر: أدوات تحويلية
(1)
The effects of climatic shifts on human
speciation are discussed in Susanne Shulz and Mark
Maslin, “Early Human Speciation, Brain Expansion and
Dispersal Influenced by African Climate Pulses,”
PLoS ONE 8
(2013): e76750. On the potential influence of Toba,
see Michael Petraglia, “The Toba Volcanic
Super-Eruption of 74000 Years Ago: Climate Change,
Environments, and Evolving Humans,” Quaternary
International 258 (2012): 1–4. On the
advantages of coastal southern Africa during this
time frame, see Curtis Marean, Miryam Bar-Matthews,
Jocelyn Bernatchez, Erich Fisher, Paul Goldberg,
Andy Herries, Zenobia Jacobs, Antonieta Jerardino,
Panagiotis Karkanas, Tom Minichillo, Peter Nilssen,
Erin Thompson, Ian Watts, and Hope Williams, “Early
Human Use of Marine Resources and Pigment in South
Africa during the Middle Pleistocene,” Nature 449 (2007):
905–908.
(2)
The tempered stone tools in question
present advantages when contrasted to the Oldowan
and Acheulean stone tools that persevered in the
human lineage for about 2.5 million years, beginning
about 2.6 million years ago. See, for instance,
Nicholas Toth and Kathy Schick, “The Oldowan: The
Tool Making of Early Hominins and Chimpanzees
Compared,” Annual Review of
Anthropology 38 (2009):
289–305.
(3)
For more on the Blombos Cave finds see,
for example, Christopher Henshilwood, Francesco
d’Errico, Karen van Niekerk, Yvan Coquinot, Zenobia
Jacobs, Stein-Erik Lauritzen, Michel Menu, and
Renata Garcia-Moreno, “A 100000-Year-Old Ochre Pro
cessing Workshop at Blombos Cave, South Africa,”
Science 334
(2011): 219–222.
(4)
Francesco d’Errico, Christopher
Henshilwood, Marian Vanhaeren, and Karen van
Niekerk, “Nassarius
krausianus Shell Beads from Blombos
Cave: Evidence for Symbolic Behaviour in the Middle
Stone Age,” Journal of Human
Evolution 48 (2005): 3–24,
10.
(5)
See Susan Carey, “Précis of the Origin
of Concepts,” Behavioral and
Brain Sciences, 34 (2011): 113–167,
159. Carey’s point is offered in response to
Karenleigh Overmann, Thomas Wynn, and Frederick
Coolidge, “The Prehistory of Number Concepts,”
Behavioral and Brain
Sciences 34 (2011): 142–144. The
authors of that piece suggest that the beads at
Blombos may have served as actual material numbers
since “a string of beads possesses inherent
characteristics that are also components of natural
number” (p. 143). In other words they suggest the
beads were the
first numbers, and that numbers were first material
and became linguistic after people labeled the
material numbers. It seems more plausible that such
valuable homogeneous items created pressures for the
innovation of linguistic numbers, a creation only
made possible because of human anatomical
characteristics. For instance, Overmann, Wynn, and
Coo lidge note that “a true numeral list emerges
when people attach labels to the various placeholder
beads” (p. 144). Such an account glosses over the
less speculative psycholinguistic evidence (see
Chapter 5) demonstrating that human adults cannot
consistently discriminate quantities of things like
beads without first using numbers. I believe the
account also underappreciates the linguistic data
demonstrating that people name numbers after hands
or fingers, not after things like beads. In short,
our hands serve as the true gateway to numbers, even
if body-external items like beads create pressures
for their creation.
(6)
The survey demonstrating a
correlation between population size and religion
is presented in Frans Roes and Michel Raymond,
“Belief in Moralizing Gods,” Evolution and Human
Behavior 24 (2003): 126–135.
My comments here are based partially on Ara
Norenzayan and Azim Shariff, “The Origin and
Evolution of Religious Prosociality,” Science 322 (2008):
58–62. The advantages of within-group
cooperation for cultural adaptive fitness,
enhanced by religion, are discussed in Scott
Atran and Joseph Henrich, “The Evolution of
Religion: How Cognitive By-Products, Adaptive
Learning Heuristics, Ritual Displays, and Group
Competition Generate Deep Commitments to
Prosocial Religions,” Biological Theory 5 (2010):
18–130.
(7)
Greek, Hebrew, Arabic, and other
languages associated with the major religions in
question have decimal-based number systems.
Therefore, the pattern being highlighted here is
likely a by-product of linguistic decimal
systems. Regardless, the pattern is also
fundamentally due to the structure of the human
hands. This point merits attention, I think,
since the profundity ascribed to some religious
numbers is not commonly recognized to be
influenced in any manner by human
anatomy.
(8)
Which is not to suggest that all
spiritually significant numbers are neatly
divisible by ten. Infact, some smaller ones are
prime numbers: there is the three of the holy
trinity or the seven deadly sins or the seven
virtues of the holy spirit or the seven days of
creation. Note that all these numbers are less
than ten. Even exceptions greater than ten are
not always as exceptional as they may seem.
Consider the importance of twelve to Islam,
Judaism, and Christianity: the twelve Imams, the
twelve tribes of Israel, and the twelve
apostles. As noted in Chapter 3, duodecimal
bases also have potential manual origins as
well.
(9)
A critical look at P values and their
history is presented in Regina Nuzzo,
“Scientific Method: Statistical Errors,”
Nature 506 (2014):
150–152.
