On Notation. Whether one believes that mathematics is created or discovered, notation is certainly created. And notation can direct the course of mathematics.

- Elisha Peterson, Unshackling Linear Algebra from Linear Notation

A common entry point into learning about the mathematics of ancient Iraq is through its notation. The sexagesimal (base 60) system written in cuneiform defines what the mathematics of ancient Iraq means to most people who have been exposed to it through textbooks and short articles. One way to get some exposure to this is to visit Wolfram Alpha, type in a number, and choose "other historical numerals" - you'll be treated to a digitized version of what Babylonian cuniform sexagesimal numbers look like.

The Wolfram Alpha implementation of its sexagesimal cuniform translator was not without its problems, but teachers are already using it in their lesson plans.

It is worthwhile to think about the long series of abstractions that have lead up to the Wolfram digitized cuneiform. As Eleanor Robson points out in her book Mathematics in Ancient Iraq: A Social History, the earliest 'writing' of numbers were actually the impressions of tokens into the clay tablets - originally the tokens were actually sealed into the clay in order to record counts. Later, it was determined that the tokens themselves could be dispensed with and that the impressions could stand in their place. Later still, the impressions became incised cuneiform markings on the tablets - one of the earlies forms of writing. For a long time since, we have been mimicking these incisions with pen and paper or typeset drawings, and now a cuneiform simulacrum can be generated automatically for us in this new digitized form.

While Robson's book tells us a lot about cuneiform writing and the various systems of ancient Iraqi mathematics, it points out that we are severely mistaken if we understand the mathematics of ancient Iraq solely in terms of notational differences between it and our more familiar ways of writing mathematics. The mathematics of ancient Iraq is not merely our familiar mathematics recast with a different base and different writing style, it is fundamentally different in ways that we have trouble appreciating. It seems that when westerners look at the mathematics of another culture or another era, we tend to view it through the fixed lense of our own mathematics, we wonder to what extent did they anticipate our current mathematics, what their "contribution" was, and in what areas they were limited. We tend to ask questions like, did they have a concept of 'zero'?, did they have a 'Pascal's Triangle?' did they know 'Pythagoras's Theorem'? This limited way of viewing the math of others leads us to make a number of untenable assumptions about the history of mathematics.

Robson's analysis quickly undermines the simple stories that are often repeated in the margins of math textbooks that place Iraq at the starting point of what became, after elaboration by the Greeks, the "Western" tradition of mathematics. As she notes early on, 'the mathematical culture of ancient Iraq was much richer, more complex, more diverse, and more human than the standard narratives allow." (p. 2) Her book attempts to provide "a new look, and a new perspective" (p. 8) on a subject that has been glossed over far to much, often from an overly simplifying western vantage.

In ancient Iraq, numeracy, literature, the mechanics of cuniform, along with the intellectual, state, and educational cultures, grew together to shape each other and to give the mathematics of this period its distinct richness. The 3000 years examined in this book show a a surprisingly diverse history of mathematical practice that begins with emergence of a rudimentary numeracy that developed into applications in accounting and engineering, and that ultimately evolved into a baroque system of numerology and mathematical divination.

It is interesting that although the period explored is an ancient one, much of it has only come to light recently. Consequently, much of the non-scholarly writing about the mathematics of ancient Iraq is quite simplistic in its outlook. Robson argues that as examples of cuneiform mathematics came to light in the early and mid 20th century, scholars gravitated towards those texts that fit with their preexisting notions of Greek and Egyptian mathematics (to the exclusion of more representative ones), a process that helped only to reinforce received orientalist ideas about the mathematics of this period.

A striking example of how our preconceptions can influence our understanding of how others understand and use mathematics is provided by contrasting two translations of an ancient cuneiform tablet (table 9.1, page 277)- an early translation from the mid 20th century is full of modern mathematical terminology, while a more recent translation attempts to be closer to the original. The translation that uses a modern lens uses the convential terms like adding, subbtracting, multiplying, and dividing numbers to describe the content of the tablet - in sharp contrast, the contextualized reading uses images of lengths "holding" each other, of "turning back" within a calculation, and of "tearing" and "accumulating" surfaces.

Robson's approach is rooted in a view of mathematics that is essentially one of social constructivism - one that she admits may not be held by the majority of working mathematicians, but that provides what may be the best perspective for historical analysis. Interestingly, she also argues that a philosophical outlook of mathematical platonism (a view held by many mathematicians and historians of mathematics) inevitably influences people to see more connections between the mathematics of different cultures than can be legitimately said to exist. If we believe that mathematical objects are "real" and "discovered" rather than invented, then we would expect different cultures to somehow discover the same mathematical truths. If we abandon this perspective, differences in mathematical practices and understandings are less surprising.

It is interesting to read about the prevelence of educational texts among the surviving cuniform tablets. Some of these tablets mimic "official" documents, but bear the tell-tale signs of being used for the education of scribes. For example, some tablets are identified as educational by "the unrealistic size of the numerical parameters, ... and the lack of credible contextual data" (p56). Creating math problems infused with realistic context has apparently plagued teachers for thousands of years. Robson suggest that there is evidence that scribal training, in some periods, made use of situated learning rather than purely rote schooling (p84).

One of the consequences of abandoning a more reductive and simplistic approach to understanding this period and its culture of mathematics is that we loose the "grand narrative" of mathematics that links ancient Mesopotamian mathematics, the Greek mathematics of Pythagoras and Euclid, and what eventually became the western tradition. The simple thread that united these mathematical traditions is cut, in favor of a much more complex weaving of influences.

Mathematics in Ancient Iraq is an important reference both for its subject and for its method. For anyone researching the history of mathematics, or this period, it provides an important resource and example. The detail and close readings of original sources and its commitment to situating these within their cultural context, essential for the method that Robson pursues, might, however, prove too much for the reader with only a casual interest in the subject. For this group of readers, hopefully the insights from this text will soon be reflected in more general works and texts. Fortunately there are many good web resources that reflect this emerging understanding of ancient Iraqi mathematics, many of them contributed or maintained by Robson herself (check out the links on her webpage).

Plimpton 322 consists of trigonometric tables in a very condensed form to economise on cuneiform space. The angles are from 30 degrees to 45 degrees, though for cotangents this gives the information for 45 degrees to 60 degrees. The great mystery as to whether the first column does or does not include 1 is explained by sec squared exceeding tan squared by 1, so this column can economically show both. Finding the square roots gives secants and tangents for which there are columns from which sines or cosecants can be calculated. We are not told this, but the other degrees can be calculated by the half angle formula cotu+cosecu equals cot u/2.

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