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In P. J. Davis and R. Hersh, The Mathematical Experience, Boston: Birkhauser, 1981.
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Mathematical Treasures: Mesopotamian Accounting Tokens
Archaeological digs in the Mideast have uncovered thousands of small clay objects, dating from as far back as 8000 BCE. These objects, referred to as “tokens”, have specific shapes and markings indicating a designated, but until recently unknown, purpose. This mystery was solved by the art historian Denise Schmandt-Besserat who began researching these items in 1969. The extraordinary results of her research were published in a number of articles and books, including How Writing Came About (University of Texas Press, 1996).
Her conclusion? The tokens were numerical counters. Their use evolved over thousands of years from simply shaped tokens (see Figure 1) to more refined incised tokens (see Figure 2). These counters were concrete, meaning each shape represented a specific quantity of a specific commodity. Employed as inventory listings, the tokens were often enclosed in clay envelopes with their imprints made on the envelope’s surface (see Figure 3). At a later time, the token imprints were made directly onto clay tablets, which then stood as accounting records (see Figure 4). This mathematical recording practice preceded the use of and, according to Schmandt-Besserat, actually led to the development of cuneiform writing.
Figure 1. The Mesopotamian accounting tokens shown above were found at Tepe Gawra, in present day Iraq, and date from about 4000 BCE. The cone, spheres, and flat disc are measures of cereals: smallest, larger, and largest. The tetrahedron designates a unit of work, perhaps one man per one day, or the amount of work performed by one man in one day. (Image provided courtesy of Denise Schmandt-Besserat and the University of Pennsylvania Museum of Archaeology and Anthropology, University of Pennsylvania, Philadelphia.)
Figure 2. Incised tokens from Tello, ancient Girsu, present day Iraq, circa 3300 BCE. Viewing the top row from right to left, the tokens represent: one length of textile, one jar of oil, uncertain, one measure of wheat. Continuing along the bottom row from left to right: one sheep, one length of rope, one ingot of metal, one garment. (Image provided courtesy of Denise Schmandt-Besserat and Musée du Louvre, Département des Antiquités Orientales.)
Figure 3. Envelope and contents from Susa, Iran, circa 3300 BCE. Each lenticular disc stands for “a flock” (perhaps 10 animals). The large cone represents a very large measure of grain; the small cones designate small measures of grain. (Image provided courtesy of Denise Schmandt-Besserat and Musée du Louvre, Département des Antiquités Orientales.)
Figure 4. Impressed and incised tablet from Godin Tepe, Iran, circa 3100 BCE. The circular imprints stand for tens and the wedges for units. The incised figure to the right is a depiction of a jar of oil, and this tablet is a record of, in total, 33 jars of oil. On this tablet, the six impressions represent abstract numbers and the incised figure a specific object. Schmandt-Besserat has argued that such impressions represent the beginning of abstract numeration and such incised pictographs the beginning of writing. (Image provided courtesy of Denise Schmandt-Besserat and the late T. Cuyler Young, Jr., of the Royal Ontario Museum, Toronto, Canada.)
For further information on the history and use of these tokens, the reader is referred to Denise Schmandt-Besserat’s How Writing Came About or one of the other sources listed below. All four of the images shown here are presented courtesy of Prof. Schmandt-Besserat, who supplied the images to Convergence, and with the cooperation of the indicated museums and/or persons. You may use these images in your classroom; all other uses require permission from both Schmandt-Besserat and the museum holding the item depicted.
For a summary, see one or both of the following articles:
Schmandt-Besserat, Denise, “Oneness, Twoness, Threeness: How Ancient Accountants Invented Numbers,” in From Five Fingers to Infinity (Frank Swetz, ed.), Open Court, Chicago, 1994, pp. 45-51.
Peterson, Ivars, “From Counting to Writing,” Science News. Originally posted Feb. 22, 1997; updated March 11, 2006.
Swetz, Frank J., "Mathematical Treasures: Mesopotamian Accounting Tokens," Loci (August 2012), DOI: 10.4169/loci003901