Counter Culture: Towards a History of Greek Numeracy

While the methodology delineated here is, I hope, sufficiently original to justify a programmatic article, it is, of course, not without its antecedents. The best treatment of the history of numeracy, in the modern context, is Porter T. , Trust in numbers (Princeton, 1995). Most directly related to this article, I mention Damerow P. Lefevre W. , Rechenstein, Experiment, Sprache (Stuttgart, 1981), an extraordinary study in the history of numerical and empirical practices. While I do not agree with Lefevre's interpretation of Greek counter culture (unfortunately, it is based on some very outdated literature), I found it of great value in developing my own thinking on the subject.

See especially Goody J. , The domestication of the savage mind (Cambridge, 1977), The logic of writing and the organization of society (Cambridge, 1986), and The interface between the written and the oral (Cambridge, 1987).

There is of course a vast field of studies in reading and writing outside the ancient world, often with interesting consequences for the history of science. One may mention — To give one important example — Stock's B. The implications of literacy: Written language and models of interpretation in the eleventh and twelfth centuries (Princeton, 1983).

The references are Cohen P. C. , A calculating people (Chicago, 1982), a study of elementary mathematical education in early nineteenth-century United States, and Brewer J. Porter R. (eds), Consumption and the world of goods (London, 1994), a study of consumerism as an historical phenomenon.

Chinapah V. , Handbook on monitoring learning achievement (Paris, 1997).

Goody J. , The interface between the written and the oral (Cambridge, 1987).

In the study of ancient literacy, an important study of this nature is Thomas R. , Literacy and orality in ancient Greece (Cambridge, 1992).

See e.g. Crosby A. W. , The measure of reality (Cambridge, 1997), chap. 6.

Ascher M. Ascher R. , Code of the Quipu (Ann Arbor, 1981).

Schmandt-Besserat D. , Before writing (Austin, 1992).

See e.g. Napier J. , Hands (Princeton, 1980).

See Dehaene S. , Numerical cognition (Cambridge, Mass., 1993), 12ff.

Pullan J. M. , The history of the abacus (New York, 1968).

Schärlig A. , Compter avec des cailloux (Lausanne, 2001).

Heath T. L. , A history of Greek mathematics (Oxford, 1921), 46–52.

I put aside the questions of other methods of calculation. Mental calculation was common, as is clear e.g. from the many papyri containing arithmetical tables: Most probably, these were meant to be memorized by heart (on tables of this kind, see Fowler D. , The mathematics of Plato's Academy (Oxford, 1999), 234 ff). Finger reckoning, a very widespread practice across many cultures, was probably practised in the Greek world, too (see e.g. Aristophanes's Vesp. 656–7, where the point may be that the calculation is so simple one can make it with one's fingers, no need for counters). As the preceding example shows, both mental and finger reckoning are confined to relatively simple calculations. They may have served as aids, to further simplify operations on the abacus, but my guess is that, mental calculating prodigies aside, complicated calculations were always done with the abacus.

As Pullan , op. cit. (ref. 13), insists, the abacus in its ancient form was also used in medieval and early modern Europe; I therefore prefer to call it “western”, to distinguish it from the eastern abacus. It also should be noted immediately that the abacus was probably not a Greek invention: I shall return to discuss its probable Near Eastern context in Section 3 below.

Lang M. L. , “Herodotus and the abacus”, Hesperia, xxvi (1957), 271–87.

Lang M. L. , “Abaci from the Athenian Agora”, Hesperia, xxxvii (1968), 241–3.

Schärlig , Compter avec des cailloux (ref. 14), chap. 3 (with references to many recent publications).

Immerwahr H. R. , “Aegina, Aphaia-Tempel, an archaic abacus from the Sanctuary of Aphaia”, Archäologischer Anzeiger, 1986, 195–214.

Lang M. L. , “The abacus and the calendar”, Hesperia, xxxiii (1964), 146–67, and “The abacus and the calendar II”, Hesperia, xxxiv (1965), 224–57.

Lang mentions some of the ancient references in “Herodotus and the abacus” (ref. 18), 271, and more can be found through Pullan, op. cit. (ref. 13), 113–14, and Schärlig, op. cit. (ref. 20), chap. 1. One should also add Epicharmus fr. 2, an important piece of evidence from a mid-fifth century comedy; and of course the entire literature for the use of counters, or pebbles, in “Pythagorean arithmetic”: We shall return to such evidence in 2.4 below.

A caveat should be mentioned: Lang was sometimes criticized in the 1960s, most notably in two publications by Pritchett W. Kendrick , “Gaming tables and IG I2 324”, Hesperia, xxxiv (1965), 131–47 and “‘Five lines’ and IG I2 324”, California studies in Classical Antiquity, i (1968), 187–215, for perhaps moving too quickly to conclude that certain artifacts were abaci and not, for instance, game boards. In this article, I try to argue that the attempt to define a sharp distinction between calculation, and other symbolic practices, is in itself misguided, so that the debate, I believe, is about a non-existent question.

Note that some use was also made in the Roman world of a more ‘eastern’ device, where buttons are moved along grooves. See Pullan , op. cit. (ref. 13), 19–20.

Fellmann F. , “Römische Rechentafeln aus Bronze”, Antike Welt, xiv (1983), 36–40. The extent to which Roman numeracy differed from its Greek counterpart is an important question I shall not enter into in this article.

The “negative position” rule, whereby values are subtracted, not added, when out of the regular decreasing order, is a modern sophistication of Roman numerals. The relation between the abacus and Latin numerals was apparently first noted by Taisbak C. M. , “Roman numerals and the abacus”, Classica et medievalia, xxvi (1965), 147–60.

Crump T. , The anthropology of numbers (Cambridge, 1990).

In many languages and in particular in all Indo-European languages, the basis of numeral words is decimal; see e.g. Dantzig T. , Number, the language of science (New York, 1930), 12.

For the following description of Greek numerical record see e.g. Heath, op. cit. (ref. 15), 26–45.

de Ste Croix G. E. M. , “Greek and Roman accounting”, in Studies in the history of accounting, ed. by Littleton A. C. Yamey B. S. (London, 1956), 56–57.

Like most numeral symbolisms of this kind, the Greek acrophonic numerals use a simple stroke for the unit.

Keynes J. M. , A treatise on money, i (London, 1930), reprinted in The collected writings of John Maynard Keynes, v (Cambridge, 1971), 3, 10.

All this has no bearing on the question of the origin of coins. Why coins were invented — Their etymology, as it were — Is a question frequently addressed in the literature, thus creating a somewhat misguided emphasis at least for the interests of this article. Here I am interested not in the question, What made people begin to make objects we identify as “coins”?, but in the question, What made such objects, eventually, so important and stable a feature of the Greek economy? The two questions are likely to get different answers.

A further clarification is necessary. It has been argued, rightly I believe, that coins came to have a central place in the Greek symbolic world (see von Reden S. , Exchange in ancient Greece (London, 1995), and especially Kurke L. , Coins, bodies, games and gold: The politics of meaning in Archaic Greece (Princeton, 1999)). This article may serve perhaps to support this thesis. Here, however, I study primarily not the symbolic world (which has to do, let us say, with what people think when they think) but the cognitive world (which has to do with what people do when they think). Regardless of what the Greeks may have thought about and through coins, they first had to be able to operate, thinking with them: It is this operability of coins I discuss here.

Fowler , op. cit. (ref. 16), 227ff. This strict adherence to unit fractions shapes the entire texture of Greek finance (taxes, shares and interest rates were all fixed with reference to unit fractions), and is thus an example of the role of the forms of numeracy in economic history.

This is based on a survey of the examples in Jenkins G. K. , Ancient Greek coins (London, 1990). It should be noted that diameters vary rather less than weights: With smaller weights, the tendency is to keep the diameter relatively constant while flattening the disk further and further.

See e.g. Howgego C. , Ancient history from coins (London, 1995), 1–8, especially the map of early minting on p. 5.

Aristophanes , Vesp. 787–93.

See e.g. Jenkins , op. cit. (ref. 37), 47, ill. 107.

Scheidel Walter , personal communication.

See e.g. Jenkins , op. cit. (ref. 37), 3. For Athens, see Kroll J. H. , The Greek coins (Princeton, 1993), ii (but note that silver-rich Attica was relatively reluctant to introduce bronze coinage: It was much more important e.g. in Sicily).

Lang M. L. Crosby M. , Weights, measures and tokens (Princeton, 1964), Part 2.

Cohen E. E. , Athenian economy and society: A banking perspective (Princeton, 1992), 69. I suspect one did not really need a separate abacus: The table itself would do equally well. The references to the abacus in the banking context may well be references to the table by another name. (To be even more speculative, this may go some way to explain this strange word for bank, “table”!).

For Babylonian sexagesimals, see e.g. Menninger K. A. , Number words and number symbols (Cambridge, Mass., 1969), 162ff. In general, for the role of mathematics in the Babylonian bureaucracy, see Robson E. , Mesopotamian mathematics, 2100–1600 B.C.: Technical constants in bureaucracy and education (Oxford, 1999).

Thus, for instance, the unit fractions on the Salamis abacus (IG II² 2777) are easily interpreted as referring to obol and drachma fractions: See Pullan , op. cit. (ref. 13), 23–24.

Note incidentally that, especially in the banking context, counters designated specially for abacus calculations were superfluous. One could simply use obols: But this is pure speculation.

There is very little research of ancient scales and their operation. Kisch B. , Scales and weights (New Haven, 1965), has some useful information. Lazzarini M. , “Le bilance romane del museo nazionale e dell antiquarium comunale di Roma”, Atti della Accademia Nazionale dei Lincei, 8 ser., iii (1948), 221–54, discusses some remarkably sophisticated Roman balances, based on the law of the balance and therefore on a truly mathematical level of numeracy.

See e.g. Lang Crosby , op. cit. (ref. 43), 2–4. I should note that I have eaten in the same Hummus joint now for almost fifteen years, and I can swear that the size of the plates has gradually diminished with time.

Lang Crosby , op. cit. (ref. 43), 34–38. In a striking metaphor of women as commodity, the stones are often transparently mastoid (perhaps the metaphor is a crude sexual elaboration of the operation of grasping the weights). Between the thumb and the finger, one held, simply, counters; other grasps called for metaphorical articulation.

Lang Crosby , op. cit. (ref. 43), 2–33.

Croix De Ste , op. cit. (ref. 31).

Hansen M. H. , The Athenian democracy in the Age of Demosthenes (Oxford, 1991), 34. An important study, Leveque P. Vidal-Naquet P. , Cleisthenes the Athenian: An essay on the representation of space and time in Greek political thought from the end of the sixth century to the death of Plato (Atlantic Highlands, N.J., 1996; transl. from a work originally publ. 1964), makes the explicit connection between early Greek reflections upon number and space, and the Cleisthenic revolution. While fully aware of the then recent scepticism of Burkert concerning “Pythagoras”, Leveque and Vidal-Naquet still posit a direct relation between Pythagoreanism and Cleisthenism. Scholars today might perhaps prefer to think of Cleisthenes not as influenced not by a specific developed doctrine, but by a more diffuse cultural practice: Numerical culture.

Compare this to modern committees, whose numbers of members is often left variable.

Political procedures may make reference to further numerical relations besides simple majority. Thus, e.g., to prevent frivolous litigation, a plaintiff in the law-courts who won less than one-fifth of the votes could face stiff penalties (once again, note the use of a unit-fraction; typically for the Athenian democracy, a simple decimal): See e.g. Hansen , op. cit. (ref. 53), 192.

Plato Rep. 522e.

Polybius IX 12–20.

For Spartan voting (of which there wasn't much taking place), see e.g. Staveley E. S. , Greek and Roman voting and elections (Ithaca, N.Y., 1972), 73ff.

de Ste. Croix G. E. M. , Class struggle in the ancient Greek world (London, 1981), 413–14. To this, one should compare the closely related conceptual nexus of the notions of coins, novelty and democracy: See the fundamental study by Kurke , op. cit. (ref. 35).

This interpretation follows Hansen , op. cit. (ref. 53), 147–8. Conflicting interpretations of Athenian show of hands were offered in the past, but Hansen's arguments against strict counts seem to me compelling.

According to another interpretation of the sources, 6000 was not the quorum for all the ostraka, but was the minimum number required against the individual to be ostracized. That our sources can be ambivalent on such a central question is in itself an interesting evidence for the history of Greek numeracy. See e.g. Lang M. L. , Ostraka (Princeton, 1990), 1–2, and references there.

Ostraka are by their very nature ready-made objects, coming in all shapes and sizes. Still, the voting ostraka fragments preserved (more than 11,000, sometimes in complete shape, and usually allowing an estimate of the original dimensions: See Lang , op. cit. (ref. 61)), are all easily graspable by hand: This was crucial, not so much for their casting as votes as for their later counting.

The dimensions of such plates were standardized in a rough way — As required by their manipulation. They were on average about 11cm long, 2cm wide and 2mm thick ( Kroll J. H. , Athenian bronze allotment plates (Cambridge, Mass., 1972), 22).

See Boegehold A. L. , The lawcourts at Athens (Princeton, 1995), 230–4, and references there.

It should be noted that all such votes were of the yes/no type, with rarely — In some trials — The complication of three or four options. Thus no writing was required. One should probably apply this reasoning in the other direction, too: The centrality of the psephoi mode of voting created a strong bias for setting up votes as simple, yes/no decisions.

I do not understand how 5n + 1 numbers were obtained by the 5n-based kleroterion. One assumes that a final “extra” was obtained through some cruder form of choice by lot.

Boegehold , The lawcourts at Athens (ref. 64), 214.

Boegehold A. L. , “Toward a study of Athenian voting procedure”, Hesperia, xxxii (1963), 366–74.

For seashells see e.g. Boegehold , The lawcourts at Athens (ref. 64), 214. Both they and pebbles are plentiful in Attica.

Ibid., 234–6.

For late law-court procedure as a whole see e.g. Hansen , op. cit. (ref. 53), 202.

Aristophanes , Wasps 94–96.

For seating and pay tokens, see Boegehold , The lawcourts of Athens (ref. 64), 67–72 and references there.

Ibid., p. viii.

In Thucydides III.49.1, where the results of the first vote on Mitylene are described, the expression is, significantly, qualitative: The sides were anchomaloi, ‘nearly equal’.

For this contrast between assembly and law-court, see Hansen M. H. , The Athenian Ecclesia (Copenhagen, 1983), 110–113.

For Spartan pre-coin, iron currency, see e.g. Hodkinson S. , Property and wealth in Classical Sparta (London, 2000), 154–76 (Hodkinson notes that foreign coins — Valued for their precious metal content — Did circulate in Sparta); for Spartan political institutions, see e.g. Andrewes A. , “The government of Classical Sparta”, in Badian E. (ed.), Ancient society and institutions: Studies presented to Victor Ehrenberg on his 75^th birthday (Oxford, 1966), 1–20.

Plato Hip. Mai. 285c.

See e.g. Parke H. W. , The oracles of Zeus (Cambridge, Mass., 1967), 129–63; Parke H. W. Wormell D. E. W. , The Delphic Oracle, ii (Oxford, 1956).

Pausanias VII.25.10.

For divination by counters in general see e.g. Halliday W. R. , Greek divination (London, 1913), 205–17.

For a discussion of this traditional picture of Pythagorean mathematics, and for a thorough deconstruction of this picture, see Fowler , op. cit. (ref. 16), 356ff.

Burkert W. , Lore and science in ancient Pythagoreanism (Cambridge, Mass., 1972).

See in particular the appendix to Fowler, op. cit. (ref. 16).

Furthermore, games, as a semiotic system, may serve as vehicles of symbolic expression. This symbolic domain is studied as such by the best modern treatment of Greek games, Kurke , op. cit. (ref. 35), chap. 7. As I have noted already in the context of coins, this article looks at cognitive, rather than symbolic questions: And I hasten to add, once again, that one set of questions does not rule out the other.

The scholarship on Greek sport (for which see e.g. Golden M. , Sport and society in ancient Greece (Cambridge, 1998)) usually does not stop to note that there was so much of it relative to other kinds of games; but its physical nature, and features such as the relative lack of quantification in ancient sport, are well understood (see e.g. ibid., 61ff).

For this probable lineage see Murray H. J. R. , A history of board-games other than chess (Oxford, 1952), chap. 6.

For ancient board games in general, see Murray , op. cit. (ref. 87), chap. 2.

Plato Grg. 450cd, Lgs 819d-820d.

See e.g. Hoffmann D. , Kultur- und Kunstgeschichte der Spielkarte (Marburg, 1995), 41ff.

Sombart W. , The quintessence of capitalism (New York, 1967, transl. of 1915).

Schmandt-Besserat , op. cit. (ref. 10).

Oppenheim O. , “On an operational device in Mesopotamian bureaucracy”, Journal of Near Eastern studies, xviii (1959), 121–8.

Michalowski P. , “Early Mesopotamian communicative systems: Art, literature and writing”, in Investigating artistic environments in the ancient Near East, ed. by Gunter A. C. (Washington, D.C., 1990), 53–70.

Murray , op. cit. (ref. 87), chap. 2.

Note that contextuality rules out technological determinism in the strong sense. An individual tool or practice does not lead, in itself, to any historical consequences. That is an appropriate point, and so I stress: I am not a technological determinist in any sense. Technology makes nothing necessary. People have ingenuity, which is precisely the ability to practise against their setting. Chess has an essential visual component; and people play blind chess. Some Greeks must have had their thumbs cut off by accident. How did they manage? Very well thank you. And yet the tools and practices available at a given culture lend themselves more easily to certain developments than to others; they thus have their role in shaping history.

For qualifications — And doubts — Concerning de St. Croix's thesis, see Macve R. H. , “Some glosses on ‘Greek and Roman Accounting’”, in Crux: Essays in Greek history presented to G. E. M. de Ste. Croix, ed. by Cartledge P. A. Harvey F. D. (London, 1985), 233–64, and Rathbone D. , Economic rationalism and rural society in third-century A.D. Egypt: The Heroninos Archive and the Appianus Estate (Cambridge, 1991). While evidence for ancient rationalism is not hard to find, it is also clear that rationalism in third-century a.d. Egypt, and in fifteenth-century a.d. Italy, led to different consequences. This, arguably, has to do with the implementation of such rationality with the aid of specific cognitive techniques.

Fowler , op. cit. (ref. 16), 222.

See Netz R. , The shaping of deduction in Greek mathematics: A study in cognitive history (Cambridge 1999), chap. 1.

See Netz R. , From problems to equations: A study in the transformation of early Mediterranean mathematics (forthcoming).