Pythagoras' Math More Than 1,000 Years Before He Was Born

From Art Daily:

Saturday November 13, 2010
Ancient Tablets Reveal Mathematical Achievements of Ancient Babylonian Culture

Tablet Plimpton 322, a table of Pythagorean triangles
a(2) + b(2) = c(2) 1,000 years before the Theorem existed

NEW YORK, NY.- An illuminating exhibition of thirteen ancient Babylonian tablets, along with supplemental documentary material, opens at New York University’s Institute for the Study of the Ancient World (ISAW) on November 12, 2010. Before Pythagoras: The Culture of Old Babylonian Mathematics reveals the highly sophisticated mathematical practice and education that flourished in Babylonia—present-day Iraq—more than 1,000 years before the time of the Greek sages Thales and Pythagoras, with whom mathematics is traditionally said to have begun.

The tablets in the exhibition, at once beautiful and enlightening, date from the Old Babylonian Period (ca. 1900–1700 BCE). They have been assembled from three important collections: the Columbia Rare Book and Manuscript Library, Columbia University; the University of Pennsylvania Museum of Archaeology and Anthropology; and the Yale Babylonian Collection, Yale University.

Before Pythagoras has been curated by Alexander Jones, ISAW Professor of the History of the Exact Sciences in Antiquity, and ISAW visiting scholar Christine Proust, historian of mathematics and ancient sciences at the Institut Méditerranéen de Recherches Avancées, in Marseille. The exhibition remains on view at ISAW through December 17, 2010.

Jennifer Chi, ISAW director for exhibitions and public programs, states, “It has long been widely recognized that many of the critical achievements of Western Civilization, including writing and the code of law that is the basis for our present-day legal system, developed in ancient Mesopotamia. However, the stunningly advanced state of mathematics in this region has largely been known only to scholars. By demonstrating the richness and sophistication of ancient Mesopotamian mathematics, Before Pythagoras adds an important dimension to the public knowledge of the history of historic cultures and attainments of present-day Iraq.”

Babylonian mathematics is known to the modern world through the work of scribes, primarily young men who, coming from wealthy families in which literacy and professional expertise were handed down through generations, were formally trained in reading and writing. Destined to work in such fields as accounting, building-project planning, and other professions in which mathematics is essential, the scribes learned and practiced mathematics by copying symbols and solving problems—some practical, others theoretical—such as those seen in the tablets in the exhibition.

Alexander Jones notes, “The evidence we have for Old Babylonian mathematics is amazing not only in its abundance, but also in its range, from basic arithmetic to really challenging problems and investigations. And since the documents are the actual manuscripts of the scribes, not copies selected and edited by later generations, we feel as if we were looking over their shoulders as they work; we can even see them getting confused and making mistakes. Recent research has made this human dimension very vivid, using archeological evidence to re-imagine the schools and the process of teaching and learning. Moreover, the contents of the tablets are still recognizable, as they continue to be taught in contemporary mathematics.”

The tablets in Before Pythagoras, inscribed in cuneiform script, cover the full spectrum of mathematical activity, from arithmetical tables copied by scribes-in-training to sophisticated work on topics that today would be classified as number theory and algebra. In so doing, they illuminate three major themes: arithmetic exploiting a notation of numbers based entirely on two basic symbols; the scribal schools of Nippur, which was the most prestigious center of scribal education; and advanced mathematical training.

Many of the problems solved by scribes at the advanced level of training were in fact much more difficult than any they would have to deal with in their careers, and their solutions depended on principles that, before the rediscovery of the Babylonian tablets, were believed to have been discovered by the Greeks of the sixth century BCE and later. One of the tablets, for example, is an extremely unusual diagram showing a square with its two diagonals and three numbers that demonstrate that what we call the Pythagorean Theorem existed 1,000 years before Pythagoras lived. The content of other tablets ranges from mathematical tables for training, to practical calculations for professionals, to abstract algebraic problems.

The meaning of these and other tablets from the Old Babylonian Period were first elucidated by mathematician and historian of science Otto Neugebauer (1899–1990), who spent some two decades, beginning in the 1920s, transcribing and interpreting tablets that had come to light in ancient Mesopotamia since the nineteenth century. It is his pioneering research, as well as the work of his associates, rivals, and successors, that revealed to modern scholars the period’s rich culture of mathematics. Before Pythagoras includes a selection of manuscripts and correspondence, on loan from the Institute for Advanced Study (Princeton, New Jersey), that offers a glimpse of Neugebauer’s methods and his central role in this “heroic age” of scientific discovery.

In order to enable visitors to appreciate the cuneiform tablets more fully, ISAW has developed an extended exhibition pamphlet that will guide viewers in reading cuneiform numbers. The exhibition features a content-rich website.