(Oxford, 2009).

OHNC

Kadosh, Roi Cohen and Ann Dowker, eds.,

The Oxford Handbook of Numerical Cognition

(Oxford, 2015).

WALS

Dryer, M. S. and M. Haspelmath, eds.,

The world atlas of language structures online

, Munich: Max Planck Digital Library, <http://wals.info>.

WWS

Daniels, Peter T. and William Bright,

The World’s Writing Systems

(Oxford, 1996).

Introduction: What is counting?

The definition quoted from Leibniz appears in the Confessio philosophi; see Wedell 2015, citing Leibniz 2005, p. 102.

It is not feasible to list here the large number of public-facing accounts of number and its history, many of which intersect with the concerns of this book. As well as those mentioned below for specific points, I have found the following of particular help in arriving at a sense of the general shape of the story (or stories) of counting: Scriba and Ellis 1969, Flegg 1983, Crump 1990, Ifrah 1998, Chrisomalis 2020.

1 Number sense before counting

The literature on innate number sense(s) is very large, and new research is constantly being published. Helpful survey collections are HMC, OHNC and Butterworth et al. 2018. Summaries of the animal evidence therein are Brannon’s chapter in HMC and Agrillo’s in OHNC. Among the many papers cited there, I have used specific evidence from Ditz and Nieder 2016 (on crows), Benson-Amram et al. 2017 (on social carnivores), Hauser et al. 2000 (p. 829 for the macaques in the Cayo Santiago), Uller et al. 2001, Cantlon and Brannon 2007a, 2007b, and Beran et al. in OHNC (on primates).

The neuroimaging evidence is discussed in Nieder’s paper in OHNC, Amalric and Dehaene 2017, Hyde 2011, Anobile et al. 2016, Roggeman et al. in OHNC, Salillas and Semenza in OHNC and Nieder 2016. The crucial finding of ‘number neurons’ was reported in papers including Nieder and Miller 2003, Nieder et al. 2004 (monkeys; see also Nieder 2013), Nieder et al. 2006, Ditz and Nieder 2015 (crows) and Kutter et al. 2018 (humans); see further Dehaene et al. in HMC, Piazza et al. 2004, Nieder and Dehaene 2009, Hyde 2011, Anobile et al. 2016, Gallistel 2017, Burr et al. 2017, Amalric and Dehaene 2017.

Neural network simulations are discussed (for instance) in Verguts and Fias 2004, Zorzi et al. in HMC, Stoianov and Zorzi 2012, Miller 2013, Hannagan et al. 2017 and Zorzi and Testolin 2017.

The evolution of number sense is discussed in Beller and Bender 2008, Hubbard et al. 2008, Nieder 2018, and Brannon and Park in OHNC.

Subitising is discussed in several of the above reviews; also in Clements 2019. An ‘impasse in the literature’ is from Agrillo in OHNC, p. 226.

2 Counting before writing

Dantzig 1930 should be mentioned as the source underlying much general writing about the history of number, including its origins in human prehistory, though much if not all of its contents are now superseded. Boyer 1944 and Scriba and Ellis 1969 cover some of the same ground; a good recent account of the foundations of number is OHHM chapter 6.1 by Chrisomalis.

Blombos: counting with beads

The Blombos beads are described in D’Errico et al. 2005; see also d’Errico et al. 2015. Other artefacts from that site are discussed for instance in Henshilwood et al. 2001, d’Errico et al. 2001, Vanhaeren et al. 2013. Other comparable artefacts are discussed in Vanhaeren et al. 2006, Bouzouggar et al. 2007, d’Errico et al. 2008, d’Errico et al. 2009; see also the review Álvarez-Fernández 2019. They have become a regular point of reference for wider discussions of embodied or enacted mathematical cognition such as Damerow 2007, Carey 2009 (chapters 4 and 7), Núñez 2009, De Cruz et al. 2010, Malafouris 2010, Overmann et al. 2011, Coolidge and Overmann 2012, Overmann 2013, 2016c, 2017a, 2017b, 2019, d’Errico et al. 2017 and Wynn et al. 2017 (related discussions of experimental evidence are in Zhang and Wang 2005 and Lindemann et al. 2007), and for more general discussions about the evolution of (‘modern’) cognition such as Balme and Morse 2006, Kuhn and Stiner 2007, Henshilwood 2007 (and other essays in Mellars et al. 2007), Botha 2008, 2016 (chapter 3), Bednarik 2008 and Jeffares 2010. (I have also used accounts that preceded this particular discovery, including Klein 2000, McBrearty and Brooks 2000, d’Errico 2003 and d’Errico et al. 2003). For the more general conceptual frameworks involved here see Kirsh 1996, Dartnall 2005, Rowlands 2010, Barrett 2011 and fundamentally Clark and Chalmers 1998, Shapiro 2019 and Malafouris 2013. The phrase ‘wild number line’ more usually appears as ‘feral number line’ in the literature, e.g. in Overmann et al. 2011, p. 143; Wynn et al. 2013, p. 128.

Lake Rutanzige to Laussel: Counting with tallies

Relevant studies of incised bones are for instance Jones 1873, Davis 1974, d’Errico and Cacho 1994, d’Errico 1995, d’Errico et al. 2001, Coinman 1996, Reese 2002, Majkić et al. 2018; see also Robinson 1993, Cain 2006, d’Errico and Cacho 1994. For the artefacts from Blombos see above, ‘counting with beads’, with Henshilwood et al. 2009; also Bullington and Leigh 2002 proposing an interpretation of the ochres as tally plaques.

For the Ishango bone see de Heinzelin 1957, 1962, Brooks and Smith 1987, Pletser and Huylebrouck 1999 and Huylebrouck 2008, 2019.

For the woman of Laussel, see Duhard 1988, 1989 in particular; and on images of women in this art more generally, often with specific reference to the Laussel carvings Duhard 1990, 1991, 1993, 1994. ‘Unfortunately, the deeper significance of this art will probably remain unknown to us’ is from http://www.musee-aquitaine-bordeaux.fr/en/laussel-venus.

Cosquer: Counting by hand

On the date of human dispersal outside Africa, Groucutt et al. 2015 is a helpful review.

The role of gesture in the origin and/or grounding of number concepts is addressed in, for example, Fischer 2012, Carey 2009, Gibson 2017 and Malafouris 2010.

The Cosquer cave is described in Clottes et al. 1992a, 1992b, Clottes and Courtin 1994, 1996, Collina-Girard 1995 and Sartoretto et al. 1995 (with details of its submergence). The second campaign of study is surveyed in Clottes et al. 2005a (on which I rely most heavily), 2005b.

Other hand stencils are described in Leroi-Gourhan 1967, Lorblanchet 1980, Groenen 1988, Barrière and Suères 1993, Delluc and Delluc 1993, Garcia and Duday 1993, Pettitt et al. 2015 and García-Diez et al. 2015. Interpretations of the hand stencils include Walsh 1979, Groenen 1990, 2011, Gilligan 2010, Cohen 2012, Culley 2016 and McCauley et al. 2018; and specifically on the possibility of a counting function Rouillon 2006 and Overmann 2014. ‘Uninvited guests’ is from Houston 2004, p. 223.

Counting words

On language origins I have used the survey collections Botha and Knight 2009a and 2009b; on the utility of mass comparison and other attempts at deep reconstruction beyond the orthodox language families I follow Campbell and Poser 2008.

Important works on language and number are Hurford 1987 and Wiese 2003, the latter proposing a purely linguistic origin for counting; see also Gelman and Butterworth 2005, Wiese 2007, Chrisomalis 2010a and Wynn et al. 2013.

The presently observable universals or near-universals of number words are documented in Greenberg 1978, Corbett 1978, Gvozdanovic 1999, Hammarström 2008a, 2008b, Donohue 2008, WALS s.v. ‘Numeral bases’ (by Comrie), Comrie 2020 and Epps et al. 2012; on ordinals, distributives, etc. see WALS s.v. ‘Ordinal Numerals’ (by Stolz and Veselinova), ‘Distributive Numerals’ and ‘Numeral Classifiers’ (both by Gil). On certain rarities and their implications for these universals see Plank 2009, Hammarström 2010.

The analogy of a bush rather than a line or ladder is from Bahn 1988, p.65.

Interlude: the numbers

A helpful way into questions about what numbers are is provided by the articles in the Stanford Encyclopedia of Philosophy (https://plato.stanford.edu) under ‘Philosophy of mathematics’, ‘fictionalism’, ‘formalism’, ‘intuitionism’, ‘nominalism’, ‘structuralism’ and ‘Platonism’. I have used from the large recent literature on this subject Badiou 2008, De Cruz and De Smedt 2010, De Cruz 2016, Giaquinto 2017 and Overmann 2016b. Allusion is also made to Benacerraf 1973.

3 Counting with words and symbols in the Fertile Crescent

Sumer: Counting symbols

Here I essentially follow Nissen et al. 1993 and Robson 2008; see also Chrisomalis 2010b (chapter 7), 2014. My account of the later development of mathematical writing is particularly indebted to Robson 2008 and to Englund 2004; see also the other chapters in Hudson and Wunsch 2004, Katz 2007 (chapter 2 by Robson) and Valério and Ferrara 2020.

Further information on the genesis of writing in this region is taken from Lieberman 1980, Walker 1987, WWS section 3, pp. 37–56 (by Michalowski and Cooper) and Overmann 2016a. I have followed what I take to be the balance of current opinion; for a different interpretation of evidence and timescales see Schmandt-Besserat 1992 and her several other publications. ‘People must have realized …’ is from Van De Mieroop’s chapter in Nissen et al. 1993, p. 61.

Wider information on the history of the region is in Kuhrt 1995, Van De Mieroop 2004 and CWP chapters 3.7 (Oates), 3.8 (Oates); for its languages see CEWAL (chapters 2 by Michaelowski and 8 by Huehnergard and Woods), and on Akkadian also Goldenberg 2013.

Tiglath-Pileser I: Counting plunder

The fundamental source for this chapter is De Odorico 1995 (see p. 125 for the city of Katmuhu), supplemented by Robson 2008 chapter 5: ‘approximation tended to be used …’ is at p. 69, ‘Typically a clean-shaven scribe’ at p. 141. Further information on the period and region is from Kuhrt 1995 (see pp. 81–108, 348–65, 473–546), Van De Mieroop 2004 (pp. 89–98, 169–74, 216–52), and also CWP chapters 3.8 (Oates; ‘as though it were a footstool’ is on p. 1503) and 10 (Özyar), pp. 1545–7. On Tiglath-Pileser I in particular, see Grayson et al. 1987, pp. 5–84; ‘I captured in battle their king’ is from p. 7, ‘I gained control over lands’ from p. 6.

Details on the Assyrian language are from CEWAL (chapter 8 by Heuhnergard and Woods) and Goldenberg 2013, and on its number words from Goetze 1946 as well as Huehnergard 2011 (pp. 235–8) and pp. 10–11, 30–31, 110 in Goldenberg 2013. Its script is described in WWS pp. 37–56 (by Cooper).