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.

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