## Mathematics in Ancient Iraq

I started Eleanor Robson’s 2008 Mathematics in Ancient Iraq: A Social History two days ago. Not my usual reading. I needed to find out what she had to say on certain issues, but the book was checked out of the library, so I decided, what the heck, I’ll just buy it and give it a try. Through the wonders of Amazon, I had a copy 48 hours later, and I’m glad.

From the book jacket:

This monumental book traces the origins and development of mathematics in the ancient Middle East, from its earliest beginnings in the fourth millennium BCE to the end of indigenous intellectual culture in the second century BCE when cuneiform writing was gradually abandoned. Eleanor Robson offers a history like no other, examining ancient mathematics within its broader social, political, economic, and religious contexts, and showing that mathematics was not just an abstract discipline for elites but a key component in ordering society and understanding the world.

The region of modern-day Iraq is uniquely rich in evidence for ancient mathematics because its prehistoric inhabitants wrote on clay tablets, many hundreds of thousands of which have been archaeologically excavated, deciphered, and translated. Drawing from these and a wealth of other textual and archaeological evidence, Robson gives an extraordinarily detailed picture of how mathematical ideas and practices were conceived, used, and taught during this period. She challenges the prevailing view that they were merely the simplistic precursors of classical Greek mathematics, and explains how the prevailing view came to be. Robson reveals the true sophistication and beauty of ancient Middle Eastern mathematics as it evolved over three thousand years, from the earliest beginnings of recorded accounting to complex mathematical astronomy.

The study of Babylonian mathematics is generally associated with Otto Neugebauer, an Austrian who began studying mathematics in the early 1920s in Göttingen, then one of the great centers of mathematics in the world. Soon his interests changed to history, and he wrote his doctoral thesis on Egyptian mathematics. The work on Babylonian mathematics that occupied him next led to the truly monumental three-volume work *Mathematische Keilschrift-Texte* (or *Mathematical Cuneiform Texts*), in which he analyzed cuneiform tablets from museums throughout the world, translating and interpreting their texts. Neugebauer’s work (partly in collaboration with Abraham Sachs; see their book *Mathematical Cuneiform Texts*) would have enormous influence on subsequent studies in the history of ancient mathematics and science. He eventually came to the US, where he worked at Brown University and then at the Institute for Advanced Study in Princeton, dying in 1990. (I must add to my ever growing list of missed opportunities that I succeeded in spending a year at the IAS, near the end of his life, without doing anything to take advantage of his presence.)

In recent years, some of Neugebauer’s analyses and conclusions have been re-examined, with new interpretations given for individual tablets and, more broadly, the purpose of the entire enterprise. This is what I have been learning about in the last week, most notably through the writings of Jens Høyrup and Eleanor Robson.

Unfortunately—missed opportunities again—I’m way too late to go back to New York and see an exhibition that was put on a little over two years ago at NYU’s Institute for the Study of the Ancient World, called Before Pythagoras: The Culture of Old Babylonian Mathematics. Here’s the website description of the show:

Since the nineteenth century, thousands of cuneiform tablets dating to the Old Babylonian Period (c. 1900-1700 BCE) have come to light at various sites in ancient Mesopotamia (modern Iraq). A significant number record mathematical tables, problems, and calculations. In the 1920s these tablets began to be systematically studied by Otto Neugebauer, who spent two decades transcribing and interpreting tablets housed in European and American museums. His labors, and those of his associates, rivals, and successors, have revealed a rich culture of mathematical practice and education that flourished more than a thousand years before the Greek sages Thales and Pythagoras with whom histories of mathematics used to begin.

This exhibition is the first to explore the world of Old Babylonian mathematics through cuneiform tablets covering the full spectrum of mathematical activity, from arithmetical tables copied out by young scribes-in-training to sophisticated work on topics that would now be classified as number theory and algebra. The pioneering research of Neugebauer and his contemporaries concentrated on the mathematical content of the advanced texts; a selection of archival manuscripts and correspondence offers a glimpse of Neugebauer’s research methods and his central role in this “heroic age.”

Edward Rothstein reviewed it in the NYT, writing that the institute

has gathered together a remarkable selection of Old Babylonian tablets from the collections of three universities — Columbia, Yale and the University of Pennsylvania — that cover a wide mathematical range. Made between 1900 and 1700 B.C., they include student exercises, word problems and calculation tables, as well as more abstract demonstrations. Under the curatorship of Alexander Jones, a professor at N.Y.U., and Christine Proust, a historian of mathematics, the tablets are used to give a quick survey of Babylonian mathematical enterprise, while also paying tribute to Neugebauer, the Austrian-born scholar who spent the last half of his career teaching at Brown University and almost single-handedly created a new discipline of study through his analysis of these neglected sources.

Only about 950 mathematically oriented tablets survived two millenniums of Babylonian history, and since their discovery, debate has raged over what they show us about that lost world. Every major history of Western mathematics written during the last 70 years has at least started to take Babylonians into account.

Rothstein even mentions Robson’s book:

In a fascinating 2008 book, “Mathematics in Ancient Iraq: A Social History” (Princeton), Eleanor Robson even suggests that many tablets like these of the second millennium B.C. “were essentially ephemera, created to aid and demonstrate recall, destined almost immediately for the recycling bin.”

But as Ms. Robson also points out, these tablets’ word problems about digging and construction, their use in teaching record keeping and calculation, and their implicit affirmation of the importance of scribes and teachers, also reveal a highly organized, bureaucratic society, an “ordered urban state, with god, king and scribe at its center.”

Among the tablets on view was Plimpton 322, the most famous of all the mathematical tablets. Here it is, as photographed by co-curator Christine Proust:

Why is it of such interest? From the website again:

Plimpton 322 reveals that the Babylonians discovered a method of finding Pythagorean triples, that is, sets of three whole numbers such that the square of one of them is the sum of the squares of the other two. By Pythagoras’ Theorem, a triangle whose three sides are proportional to a Pythagorean triple is a right-angled triangle. Right-angled triangles with sides proportional to the simplest Pythagorean triples turn up frequently in Babylonian problem texts; but if this tablet had not come to light, we would have had no reason to suspect that a general method capable of generating an unlimited number of distinct Pythagorean triples was known a millennium and a half before Euclid.

Plimpton 322 has excited much debate centering on two questions. First, what was the method by which the numbers in the table were calculated? And secondly, what were the purpose and the intellectual context of the tablet?

I sure wish I took an interest in this subject three years ago. I could have read Robson’s book, then arranged for us to be in New York during the exhibition.

Anyway, I’m reading the book now, and Robson concludes (I don’t usually jump to the end, but I did this time) with a compelling argument for taking an interest in the subject:

Compared to the difficulties of grappling with fragmentary and meagre

nth-generation sources from other ancient cultures the cuneiform evidence is concrete, immediate, and richly contextualised. We can often name and date individuals precisely; we have their autograph manuscripts, their libraries and household objects. This opens a unique window onto the material, social, and intellectual world of the mathematics of ancient Iraq that historians of other ancient cultures can only dream of.