Evaluating Ancient Numeracy: Social versus Developmental Perspectives on Ancient Mesopotamian Numeration

Evaluating Ancient Numeracy:

Social versus Developmental Explanations of Archaic Mesopotamian Numeration Stephen Chrisomalis

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Cite as: Chrisomalis, Stephen. 2005. Evaluating Ancient Numeracy: Social versus Developmental Perspectives on Ancient Mesopotamian Numeration. Paper presented at the Annual Meeting of the Jean Piaget Society (Vancouver, British Columbia).

One of the contributions of anthropology to developmental psychology is the association of

cognitive processes relating to quantity with the external representational systems for numbers used in

non-Western societies. Recent research in Amazonian anthropology among the Piraha, who apparently

lack verbal numerals and have tremendous difficulty in achieving even a minimal level of numeracy,

suggests that there is an important correlation. Yet in this case and others - for instance, the work of

Geoffrey Saxe or David Lancy in Papua New Guinea - researchers have access not only to the

representational systems used, but also to experimental and ethnographic evidence. When evaluating

the numeracy of past peoples, however, we have only material remains. This is a challenge to scholars

hoping to reach inside the minds of long-dead informants in order to correlate observable these remains

- most frequently written numerals - with unknown mental processes. As an anthropologist studying

numeracy, working with historical and archaeological, rather than ethnographic data, I will address this

issue today with reference to the earliest accounting documents of ancient Mesopotamia.

The so-called 'proto-cuneiform' texts are clay tablets impressed with signs, primarily discovered

at the site of Uruk, approximately 250 km south of Baghdad on the Euphrates River; the name of the

country of Iraq derives from Uruk. Between 3300 and 3000 BC, Uruk was a walled city-state with a

population of approximately 50,000. The texts of this period of Mesopotamian history, often called the

archaic period, represent a protohistoric system of bookkeeping and administration, which is

antecedent to phonetic writing, but does not record language. Nearly six thousand clay tablets, mainly

accounting records, have been recovered which record this notation, over five thousand of them from

Uruk itself. By far the most salient feature of proto-cuneiform texts are around 60 graphic signs for

numbers (numerical notation, as opposed to number words) written upon them.



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The proto-cuneiform numerals were deciphered in the 1980s by Assyriologists Hans Nissen and

Robert Englund along with developmental psychologist Peter Damerow. They established that the sixty

numerical signs from Uruk constituted no fewer than fifteen different numerical notation systems. Each

system was used exclusively for enumerating a specific category; for instance, here we see the regular

base-60 or sexagesimal system, used for enumerating almost all discrete objects, including humans,

another used to denote capacity measures of grain, and one used to denote measures of land area. They

conclude from this feature that the numeral-signs do not represent abstract numbers, but instead are

context-dependent, representing only concrete quantities of particular types of objects rather than 'pure'

or abstract numbers.

Another odd feature is that identifying the specific numerical value the signs is difficult. For the

main system used for counting discrete objects, it is easy to identify the basic sign for 1 (since, for

instance, fractions of humans do not normally occur in texts). For systems that measure area or

capacity, however, we can never ascertain with certainty which sign (if any) has the basic value of one

unit. For this reason, we describe the archaic Mesopotamian systems using factor diagrams. We can

identify the ratio between two signs, as shown by the numbers above the arrows in this diagram, but not

specific numerical values. This ambiguity is exacerbated because the value of any particular sign - for

instance, the large circle - may vary from system to system. Here, we see that the large circle has a value

of 3600 of the basic wedge for 1 in the sexagesimal system at the top; however, it is only 60 times the

basic wedge in the in the middle system, and 1080 times the basic wedge in the one at the bottom. The

signs are thus polyvalent, and their meaning depends on the system in which they were used.

Damerow, as a developmentalist, has taken this argument the furthest in his work combining

Piagetian insights with archaeological evidence. For him, the archaic numerals represent the third in a

five-stage historical stage-based sequence in the evolution of the number concept, beginning with an

unsystematic preoperative stage, and followed by the proto-arithmetic used in the Near East starting



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around 10,000 BC. Due to time constraints, I am unable to discuss the work of Denise Schmandt-

Besserat, the discoverer of the pre-literate Mesopotamian token accounting system, who also takes a

developmentalist approach to Mesopotamian numeracy. Whereas Damerow's work is based in Piaget

and Vygotsky, Schmandt-Besserat's hypotheses are based solely on the work of Lévy-Bruhl, and are not

supported by even speculative evidence.

Damerow's hypothesis is intuitively appealing, either for a developmentalist working from a

social or constructivist position, or for an anthropologist interested in the development of mental

concepts. Yet I think there are several reasons to doubt its specific claims. While there are

developmental correlates of numerals, they do not proceed in a rigid stage-based manner, nor are the

effects of context-dependence and polyvalence so great as to prohibit the development of operational

thought. An examination of the social factors and numerate practices of archaic Mesopotamia suggest

that there were no insurmountable barriers to a fairly complex and, I suspect, formal-operational

concept of number among the small sub-section of society at Uruk that was trained in the manufacture

and reading of these texts.

An initial difficulty with Damerow's theory is that the universal stages he postulates did not exist

in other, unrelated societies that independently invented numerical notation. If this developmental

sequence were universally valid, we would expect that in ancient Egypt, in the ancient Shang Dynasty of

China, among the Maya of Mesoamerica and the Inka of Peru, that everywhere, polyvalent, multiple

numerical notation systems would exist just prior to the development of single, monovalent systems.

Here I depict several other systems that I have studied in my comparative research on numerical

notation. Nowhere else in the world does numerical notation go through a stage of context-dependent

or polyvalent numerical notation. This suggests, at the very least, that what may have been true in

Mesopotamia was not true elsewhere.



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Yet there are reasons to doubt its relevance even in Mesopotamia. One source of evidence

popular among developmentalists from the time of Levy-Bruhl comes not from symbolic number-signs

but from the numeral-words used in a given society. Lévy-Bruhl noted that many languages use several

series of context-dependent numeral-words, which classify the type of thing being counted as well as

quantifying it. These numeral classifier systems are in fact very widespread worldwide. Here we see the

system used by the Tsimshian of British Columbia. The analogy between these context-dependent

systems and the proto-cuneiform numerals is obvious. Yet their cognitive effects are by no means clear-

cut. Brent Berlin studied the numeral classifiers of the modern Tzeltal Maya of Guatemala without

finding any stage-like effects of numeral classifiers. The modern Mandarin and Japanese languages have

numeral-classifiers for counting different kinds of objects - but arguably children are more numerate in

those societies than their age-mates who speak Indo-European languages. Classifier systems are really

no different from grammatical gender - they alter words, but French speakers understand perfectly well

that le chat and la chatte are both cats, all the same. The archaic accountants of Uruk probably spoke

Sumerian. As can be seen in this chart, the Sumerian lexical numerals are sexagesimal, like the

numerical notation, but there is no evidence of a numeral classifier system, only a single context-

independent set of numerals. If Damerow is right and polyvalence and context-dependence imply an

absence of abstract number concepts, then paradoxically, the quasi-literate Uruk accountants would be

less numerate than the average Sumerian who did not use texts, only number words.

I suggest that if there are representational effects of numerals, they are most likely to resemble

Piagetian stages only in languages that lack a numerical base. Most languages, including Sumerian, have

such a base - a number whose powers specially structure the set of numerals. In base-structured

systems, phrases such as 'six hundred' exist - multiplicative, meta-representational phrases; they do not

count objects, but count counts. The existence of a Sumerian word 'six hundred' demonstrates that the

Uruk accountants knew what a 'hundred' is and could count them. Conversely, experimental and



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ethnographic advice suggests that speakers of languages with few numeral words - e.g., the Tauade

studied by Hallpike, or the Piraha studied by Gordon and Everett - do, in fact, have those difficulties.

There is no reason, nevertheless, to agree with the Whorfian linguistic relativity hypothesis that the lack

of numerals causes this lack of abstraction. It is more likely that a society possessing significant social

needs for numerate practices - will inevitably develop base-structured numerals, and in some cases, also

numerical notation.

There are sensible reasons why someone with an abstract number concept would use multiple

systems for representing numbers. We do it all the time, because of course we have both lexical numeral

words and graphic number symbols. We can even distinguish written and verbal lexical numerals. We

use Roman numerals to distinguish the foreword of a book from its main text because it is useful to have

two sets of numbers to avoid confusion and ambiguity. We use sexagesimal or base-60 numeration for

notating time (2:15, as a time notation, is very different structurally from the number 215) because our

time-measuring system is derived from Babylonian astronomy. We use binary or hexadecimal numerals

for because it is useful to have a numerical base that is a power of 2 for electronics. We use scientific

notation for expressing large numbers in a brief and readable way. Ironically, one of the principles

behind the 'new mathematics' movement in North America in the 1960s was to teach multiple,

differently-structured numerical systems to improve students' understanding of abstract number

concepts. If the archaic systems arose out of a desire to match numeration systems with existing

metrological systems, the Uruk accountants may realized that context-independent written numerals

were not the most efficient solution to the technical problems they were facing.

Unfortunately, we have little evidence for the social contexts of the use of numerals. We have no

idea how many of these systems would have been known to any individual scribe, and no evidence from

the archaic period as to how numerals were manipulated and used arithmetically. We do not know what

computational technologies were used - whether, for instance, there was a sort of abacus used at Uruk.



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The documents do not provide actual calculations - the numerals in the texts were not being lined up

and worked with arithmetically, but record results of what must often have been complex calculations

(including both whole numbers and fractions). We simply have values and totals, which tell us little

about how people were actually thinking about number. I am thus reluctant to endorse Damerow's

hypothesis that the Uruk accountants, who must have spent the bulk of their time working with

numbers, had number concepts similar to those of six to eleven-year-old Western children.

There is no reason to postulate a close correlation between forms of numeration and

developmental stages. All the ancient Greek mathematical achievements took place without the

presence of a positional or place-value numerical notation system. Medieval Western logicians from St.

Augustine to St. Anselm used only Roman numerals, but we do not think that they were incapable of

operational thought regarding number or any other domain. The experimental evidence undertaken to

date - e.g., Miller's work with Chinese and English-speaking children, or Stigler's work with abacus-

trained Chinese children - suggest that representational effects do exist - but they are not as significant

as Damerow would have us believe - certainly not approaching the level of five distinct stages. The myth

of the innumerate accountant cannot be sustained.

Around 2900 BC, there was a marked decline in the frequency of almost all the proto-cuneiform

numerical systems, while the sexagesimal system rapidly assumed the functions of the other systems.

While each metrological system had its own numerical notation system in the archaic period, eventually

all quantities were expressed using a single notation: the ordinary sexagesimal system used in the

archaic period for discrete objects only.

Why did this occur? Firstly, perhaps the use of so many systems in so many different functions

was cumbersome for administration and open to abuse, and was abandoned because it was perceived to

be so. This may, however, simply be a modern Western prejudice. Secondly, while the archaic texts were

used at only a very few locales (mainly at Uruk), the later numerals were used throughout Mesopotamia.



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The use of a single system to facilitate communication among more people would be advantageous.

Thirdly, changes in Sumerian measurement systems may have reduced the usefulness of the proto-

cuneiform systems by eliminating the fit between metrology and numeration. Nissen, Damerow and

Englund demonstrate conclusively that the multiple archaic systems correspond very well with

differently-structured systems for measuring and capacity.

The fourth explanation, Damerow's explanation, holds that the decline from many to one system

represents a cognitive change in its users - the shift from his third stage of 'archaic arithmetic' to the

fourth, 'primary arithmetic'. Damerow holds the development of a context-independent monovalent

symbolic numeration represents a step towards abstraction - though not the last step, the abstract

number concept, which would have to wait for the Greeks. While it is impossible to disprove this

explanation, I do not think it is very likely. Even if there are cognitive changes associated with the shift

from context-dependent systems with polyvalent signs to a context-independent system with

monovalent signs, any cognitive changes associated with representational systems must have occurred

in conjunction with social and technological changes. If there are epigenetic explanations for cognitive

change, they must always incorporate external, social factors or else they become teleological and

unhistorical.

In conclusion, we know far too little about the numeracy and number-related practices of proto-

literate Mesopotamians to assert that they had no abstract concept of number, or that they were

working at a concrete-operational or pre-operational level. What we do know suggests that they had a

context-independent, monovalent set of number words with a numerical base in which multiplication

figures prominently. The existence of so many accounting documents from Uruk, far more than from

any comparable ancient society, demonstrates the tremendous social importance of numerate practices

at Uruk. The evidence cannot reasonably be interpreted as representing a unilinear progression from

concreteness to abstraction. There is no simple correlation between means of visual representation and



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adult numerical abilities. Examining the social context in which numerals are used suggests instead that

there are very sensible reasons why the archaic Mesopotamians would use multiple systems. These

parallel the very same reasons why we use multiple representational systems for number depending on

context.

In closing, I leave you with a visual reflection that I hope demonstrates that polyvalence in

symbolic systems for numeration does not imply the absence of abstraction ...

There are 10 kinds of people; those who understand binary and those who do not.

Thank you.

Evaluating Ancient Numeracy Social versus Developmental Explanations of

Archaic Mesopotamian Numeration

Stephen Chrisomalis

Dept. of Anthropology, University of Toronto

Mediterranean

Sea

Egypt

Iran Baghdad

Iraq

Turkey

Saudi Arabia

Uruk Proto-Cuneiform Tablets

Uruk Tablet MS1717 (31st century BC, Uruk); Source: Schøyen Collection)

Proto-Cuneiform Systems

Context-Dependence

Proto-Cuneiform Systems

Polyvalence

Damerow’s Stages of Numeracy

(Damerow 1996: 296)

Token Accounting

Ancient Numeration

Numeral Classifiers (adapted from Lévy-Bruhl 1926: 195)

Sumerian Numerals 1

diš

30

ušu

3x10

2

min

40

nimin

20x2

3

eš

50

ninnu

20x2+10

4

limmu

60

geš

5

ia

70

geš-u

60+10

6

aš

80

geš-niš

60+20

7

imin

...

8

ussu

120

geš-min

60x2

9

ilimmu

180

geš-eš

60x3

10

u

...

11

u-diš

10+1

600

geš-u

60x10

12

u-min

10+2

1200

geš-u-min

60x10x2

...

...

1800

geš-u-eš

60x10x3

20

niš

3600

šar

21

niš-diš

20+1

7200

šar-min

3600x2

(Adapted from Powell 1971: 47-80)

Multiple Representations

three hundred and sixty-five

trois cent soixante-cinq

[Tr r

[rw ) w)

365

CCCLXV

16D

101101101

3.65 x 102

Written Verbal

L

E

X

I

C

A

L

S

Y

M

B

O

L

I

C

Early Dynastic Numeration (2900 BC – 2350 BC)

-Monovalent numerals

-Single, context-

independent system

-Damerow: Stage 4,

‘primary arithmetic’ –

universal numeration

but no conception of

numbers as ideal

objects

There are 10 kinds of people:

those who understand binary

and those who do not.

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Evaluating Ancient Numeracy: Social versus Developmental Perspectives on Ancient Mesopotamian Numeration