December 3, 2015
"From about 6000 BCE, long before writing, Neolithic villagers used simple geometric counters in clay and stone to record exchange transactions, funded by agricultural surpluses. As societies and economies grew in size and complexity, ever more strain was placed on trust and memory. By the late fourth millennium the intricacies of institutional management necessitated both an increasing numerical sophistication and the invention of written signs for the commodities, agents, and actions involved in controlling them." (27) "Mesopotamian city states had implemented an extensible and powerful literate technology for the quantitative control and management of their assets and labour force. In doing so, they had created in parallel a new social class---in Uruk called the umbisag 'accountant/scribe'---who was neither economically productive nor political powerful, but whose role was to manage the primary producers on the elite's behalf." (40)
"It was the ... state bureaucracy in which the scribes were embedded that ... drove the need for ... the sexagesimal place value system ... by imposing increasingly high calculational standards on its functionaries through the demand for complex annual balanced accounts." (83) This went hand in hand with "centrally imposed reforms of weights and measures throughout the third millennium" (84). "None of these newly invented units of measure was recorded with compound metrological numerals, but always written as numbers recorded according to the discrete notation system followed by a separate sign for the metrological unit," (76) unlike the earliest sources, where, "while there was a single word for 'ten'," "there was no single numeral but different signs for 'ten-discrete-objects', 'ten-units-of-grain', and 'ten-units-of-land'" (33).
"In the early Old Babylonian period [c. 1850 BCE], elementary scribal training underwent a revolution ... in which emphasis was more on the ability to manipulate imaginary lines and areas in almost algebraic ways than on the ability to count livestock or calculate work rates." (86) "Topics range from apparently abstract 'naive-geometrical algebra', via plane geometry, to practical pretexts for setting a problem---whether agricultural labour, land inheritance, or metrological conversions. Even the most abstract problems may be dressed up with 'practical' scenarios." (89)
This was "a style of mathematics that encapsulated the principles of ... justice" on which the society was based; "in solving abstruse puzzles about measured space, the true scribe demonstrated his or her technical capability ... for upholding justice and maintaining social and political stability on behalf of king and god" (266).
As one scribe put it: "When I go to divide a plot, I can divide it; when I go to apportion a field, I can apportion the pieces, so that when wronged men have a quarrel I soothe their hearts ... Brother will be at peace with brother." (122)
"The Sumerian word for justice was nig-si-sa, literally 'straightness, equality, squareness', Akkadian misarum 'means of making straight'. The royal regalia of justice were the measuring rod and rope ... In [this] light ... Old Babylonian mathematics, with its twin preoccupation of land and labour management on the one hand and cut-and-paste geometrical algebra on the other, becomes truly comprehensible." (123-124)
This tradition effectively came to an end with "the collapse of the Old Babylonian kingdom in c. 1600 BCE" (151), though "traces of Old Babylonian mathematical learning lingered on long after the political ideology that it supported had disappeared" (181). "Evidence suggests that mathematics ... was still a vital component of Babylonian intellectual life" (151) for a while, but ultimately a massive decline followed. "In the first half of the first millennium we find a low level of mathematical sophistication in school, consumer, and professional contexts." (212)
"Mathematics and mathematical astronomy were central components of the last flowering of cuneiform culture." (261) "From the mid-seventh century [BCE] onwards, ... compilers of eclipse records and astronomical diaries had begun to think in terms of divine quantification. ... Apparently random events of great ominous significance were observed, quantified, and recorded in the hope that numerical patterns could be detected amongst them. The ultimate aim was to understand the will of the gods, to ensure that they were propitiated and would act benignly to the king and humanity. Thus in later Babylonia mathematics became a priestly concern." (268)
These priests "comprised a tiny number of individuals from a restricted social circle, intermarrying, working closely together to train each successive generation, and highly valuing privacy and secrecy" (261-262). "Their sole aim was to uphold the belief systems and religious practices of ancient times." "In this context [they] developed increasingly mathematically sophisticated means to ensure the calendrical accuracy of their rituals." (262)