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).
Teianti: Counting coins
For the history of Egypt in this period I rely mainly on Bagnall 1993, CWP chapters 1.16 (Hendrickx) and 1.17 (Ikram), and Erskine 2003 chapters 7 (Thompson) and 15 (Rowlandson); see also Rowlandson 1998. Language and writing are surveyed respectively in CEWAL (chapter 7 by Loprieno) and WWS section 4, pp. 73–83 (by Ritner) (see also Millet 1990 and Dreyer 1992); number and mathematics in Katz 2007 (chapter 1 by Imhausen), Imhausen 2016 (‘woe to your limbs’ is on p. 144), and Chrisomalis 2010b (chapter 2).
The document discussed here is reproduced, transcribed and translated in Glanville 1939, pp. 43–5 (the quotation is from page 43); discussions of the archive are in Pestman 1989 (pp. 14–23) and at greater length in Depauw 2000. See also https://www.trismegistos.org, record 43769, with a different interpretation of the date.
4
Counter culture from Athens to the Atlantic
Philokleon: Counting votes
The general line of thought in this section depends heavily on Netz 2002, whence the phrase ‘counter culture’ (for a broadly similar take on Roman – specifically Pompeiian – materials, see Bailey 2013). Further discussions of ancient Greek numeracy are in OHHM (chapter 2.1 by Asper) and Cuomo 2001, 2012, 2019 (see also van Berkel 2017 and Sing et al. 2021).
Greek number words are discussed in Keyser 2015; see further for the Proto-Indo-European counting words Szemerency 1960, Gvozdanović 1992, Mallory and Adams 2006, pp. 61–2 and Beekes and de Vaan 2011 (chapter 16). Greek rhetorical uses of number are surveyed in Hawke 2008; see also Rubincam 1991, 2001, 2003. The quotation from Demosthenes (27.9–11) appears in Schaps 2004, pp. 18–19.
The Greek number notations are discussed in Tod 1911–12, 1913, 1926–7, 1936–7, 1950, 1954 (all reprinted in Tod 1979) which most subsequent discussions take as a starting point. Recent contributions include Verdan 2017 and Keyser 2018; for the Roman notation Keyser 1988 is also important. Crucially, I follow Chrisomalis 2003 on the relationship of the Greek alphabetic numerals to Egyptian models; see further Chrisomalis 2010b (pp. 140–43).
Athenian courts are discussed in Boegehold 1995, and the exchange of counters therein in Boegehold 1963; the kleroterion is also detailed in Bishop 1970, Staveley 1972 and Hansen 1991. Many of the details of court procedure presented here for Philokleon’s day may be found in Aristotle’s Athenian Constitution; his name is from Aristophanes, Wasps. Other voting practices such as those at the Assembly are detailed in Kroll 1972, Staveley 1972, Lang 1990 and Hansen 1991.
From the immense literature on Greek coins I have consulted Price 1983, Howgego 1995 (chapter 1), von Reden 1997 and Schaps 2004; see also Haselgrove and Haselgrove and Krmnicek 2012. The lines about ‘figs, marshals of the court …’ appear in Schaps 2004, p. 113, quoting Eubulus via Athenaeus XIV 640b–c. On Athenian and wider Greek accounting I have used the essays by Davies, Harris, Kallet-Marx and Langdon in Osborne and Hornblower 1994, and also Cuomo 2013.
Marcus Aurelius: Counting years
The anecdote about Marcus Aurelius is from Cassius Dio, Roman History 71.32, translated by Earnest Cary. Bede’s line ‘very useful and easy’ is translated in Wallis 1999, p. 9. The Latin system is discussed in Froehner 1884, Friedlein 1869, Bechtel 1909, Richardson 1916, Cordoliani 1942, 1961, Hilton Turner 1951 (p. 69), Marrou 1958, Alföldi-Rosenbaum 1971, Williams and Williams 1995, Minaud 2006, Weddell 2012 and BNP s.v. ‘digitorum computatus’; Bede’s account in particular in Jones 1937, 1943 (pp. 329–31) and Cordoliani 1948. On Greek appearances see Schöll 1830 (pp. 345–7), Liddell and Scott 1996 s.v. ‘πεμπάζω’, Marrou 1958 and Arias and Hirmer 1960, pp. 41–2, 54–5; on Rabanus Maurus see Tannery 1886 and Stevens 1968. For the system in Arabic contexts see Rödiger 1846, Palmer 1869, Lemoine 1932, Welborn 1932, Saidan 1978, pp. 349–50.
‘Give me a thousand kisses’ is from Catullus 5, 7–8, on which see Minaud 2006; for a counting-board interpretation of the passage see Levy 1941; also (and for the ‘impudicus’ in another context) Turner 1951.
Blanche of Castile: Counting with silver
Still definitive on the counting board in general, and on jetons, is Barnard 1916 (see also Jack 1967). I have also used Sugden 1981, Schärlig 2006 and as well as OCD and BNP s.v. ‘abacus’. Greek counting boards are discussed in the latter as well as in Liddell and Scott 1996 s.v. ‘ἄβαξ’, and in Schärlig 2001a, 2001b. Controversy about the identity of certain claimed Greek abaci may be seen in Lang 1957, 1964, 1968 and Pritchett 1965, 1968. ‘More a state of mind than an artefact’ is from Netz 2002, p. 327.
For the possibility of Mesopotamian counting boards see Woods and Feliu 2017. Roman counting boards are discussed in Levy 1941, Wyatt 1964 and Fellman 1983 as well as in Schärlig 2006; Roman accounting in Scheidel 1996, Cuomo 2011 and Bailey 2013.
For the medieval counting board, Barnard 1916 may be supplemented by Haskins 1912, Baxter 1989, Schärlig 2003 and Periton 2015. I have also used information from the Oxford English Dictionary s.vv. ‘chess’ and ‘exchequer’. My account of Köbel 1514 is based directly on that book itself. Chaucer’s ‘counting-bord’ appears in ‘The Shipman’s Tale’, line 83; the Shakespearean insult ‘counter-caster’ in Othello 1.1.31.
More specifically on jetons, I have also used Turckheim-Pey 1997, Barnard 1920, Smith 1921, Freeman 1946, McLellan 2020 and Sarmant and Ploton-Nicollet 2010–.
On Blanche of Castile I have relied mainly on Grant 2016. ‘Remarkably adept at ensuring the people did what she wanted’ is from p. 11, ‘She reacted courageously to challenge’ p. 104, ‘fashion and charm’ p. 319. Her accounts are also discussed in Bougenot 1889.
Number symbols from India
On the name of this set of symbols see Chrisomalis 2010b, p. 1 note 1; I have tried within reason to call the Brahmi/Indian/Toledan/Arabic numerals what the people in my stories called them at the time.
Bhaskara II: Brahmi numerals
For the wider context of writing in South Asia, see WWS section 30 (by Salomon) and CEWAL (chapter 26 by Jamison). On the Arabic numerals in particular, Smith and Karpinski 1911 and Ginsburg 1917 are still valuable; Chrisomalis 2010b (chapter 6) is my main source.
For the context of mathematics in India I have used Katz 2007 (chapter 4 by Plofker) and Plofker 2009. ‘You, radiant Agni’ is quoted in Plofker 2009, p. 13, from Rg-veda 2.1.8, ‘Hail to a hundred’ at p. 14, from Yajur-veda 7.2.20; for ‘the universe is created and destroyed’ see p. 53. The Lilavati is discussed on pp. 182–91 and elsewhere; see also Plofker 2009b. ‘Oh Lilavati, intelligent girl’, ‘Tell me, quick-eyed girl’ and ‘There is a hole’ are from Katz 2007 (chapter 4 by Plofker), pp. 449, 454 and 459; ‘A traveler on a pilgrimage’ from Plofker 2009, p. 185.
Counting words in Sanskrit and its descendants are detailed in CEWAL (chapter 26 by Jamison).
Ibn Mun’im: Dust numerals
On the period and location I have used information from Bennison 2016, Abun-Nasr 1987 and Fleet et al., s.v. ‘Almohads’; on the sciences in the Arabic-speaking world from Katz 2016 (chapter 3 by Berggren), Brentjes in OHHM, 2018. A system ‘which calls for no materials’ is from Kunitzsch 2003, p. 5, from Muhammad ibn Yahya al-Sali, Adab al-kuttab; ‘misbehaved’ people from Katz 2016 (chapter 3 by Berggren), p. 532.
Number symbols are discussed in EHAS s.v. ‘Numeration and arithmetic’ (by Saidan), Kunitzsch 2003 and Burnett 2002a, 2005, 2006; see also the classic paper Gandz 1931. ‘Subtle discoveries’ is from Burnett 2006, 15; ‘the Indians have a most subtle understanding’ from p. 17 with note 4 (translation modified).
Ibn Mun’im’s combinatorics is discussed in Djebbar 1985, with a French translation of the text. See also Djebbar n.d., EHAS s.v. ‘Numeration and arithmetic’ (by Saidan), Katz 2007 (chapter 5 by Berggren), pp. 660–5, and Katz 2016 (chapter 3 by Berggren), pp. 427–46.
