I knew about Plimpton 322 for many years. I had followed the presentation by B. L. van der Waerden in Science Awakening, Oxford University Press, 1971, and was familiar with that work. Not until recently did I return to a keen interest in Old Babylonian mathematics. When I read the paper by Eleanor Robson, Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322, published in Historia Mathematica, Vol 28, pages 167 - 206, 2001, I became aware of more recent work on the famous tablet. Robson, who is regarded as an authority on Mesopotamian mathematics at the University of Cambridge, has made the case for a more mundane solution, arguing that the tablet was created as a teacher's aid, designed for generating problems involving right triangles and reciprocal pairs. See also Words and Pictures: New Light on Plimpton 322, American Mathematical Monthly 109 (2002), 105-120.
I shall make detailed reference to the Sherlock Holmes paper; the reader should know the arguments presented therein.
See also remarks about Plimpton 322 from the Columbia University website:
Cuneiform Tablet. Larsa (Tell Senkereh), Iraq, ca. 1820-1762 BC. -- RBML, Plimpton Cuneiform 322. A gift of George Arthur Plimpton, 1936.
The clay tablet with the catalog number 322 in the G. A. Plimpton Collection at Columbia University may be the most famous mathematical tablet from ancient times. It was scribed in the Old Babylonian period and shows the most advanced mathematics, more than a thousand years before the development of Greek mathematics. It is unique. No other tablets like it have been found, in contrast to many "school" tablets of reciprocals and mathematical problems found in abundance.
Plimpton 322 is known throughout the world to those interested in the history of mathematics as a result of the work of Otto Neugebauer, chair of Brown University's History of Mathematics Department. In the early 1940's he and his assistant Abraham Sachs discovered that it contained Pythagorean triples, integer solutions of the equation a2 + b2 = c2.
This is a photograph of the famous tablet:
The tablet comes from the same period as the original writing of an Egyptian mathematical papyrus, attributed to the reign of Amenemhet III, circa 1800 BC. (The Rhind Papyrus actually dates from the reign of the Semitic Hyksos King Apophis, circa 1550 BC, but references that earlier date.) Marshall Clagett, in Volume Three, Ancient Egyptian Mathematics, American Philosophical Society, Philadelphia, 1999, remarks about the possible influence of Babylonian mathematics on Egyptian developments. Clagett puts the original text (of the Plimpton tablet) squarely in the very fertile period of the (Egyptian) development of mathematical tables and problem collection. Thus it appears that both Egypt and Mesopotamia were in heavy mathematical activity at the same time.
As I have shown, there was considerable cultural influence on the Egyptians by the Mesopotamians. See
Hence it seems safe to deduce that the Egyptians were aware of the use of applied mathematics in Mesopotamia. The laying out of fields, digging of canals and ditches, computing manpower necessary to build structures, and so on, were similar in both regions. Hoyrup (reference below) mentions the widespread dispersal of knowledge in the Mesopotamian region on pg 349 of his work: YBC 4662-63 offer evidence that problems did not only travel between schools and regions but were systematically borrowed and adapted to the local canon. A view that Egypt and Mesopotamia were both isolated from one another in cultural cross fertilization is a very narrow notion of the world's ancient past. Unfortunately, this monodomous view of ancient cultures is widespread throughout modern scholarship. Egyptian scholars know little about Mesopotamia, while Mesopotamian scholars are in equal ignorance about Egypt.
We also know that considerable commercial exchange went on before 4,000 BC, illustrated by Egyptian lapis lazuli, obtained out of the mountains of Afghanistan.
When we consider the past history of mankind, with unresolved impact of one culture upon another, we must take our perspective beyond that of the immediate historical context. What were the social elements that led to a period of profound simultaneous mathematical explorations in both Egypt and Mesopotamia? Did cross fertilization yield higher thinking in both cultures? Or is it possible that still other planetary elements contributed to this fruitful intellectual period?
Consider, for example, the geological history of our world. About ten thousand years ago a great planetary meteorological change occurred. The ice caps suddenly melted leading to widespread dispersal of populations in animals and in man. This phenomenon took place all over the globe. The mountains of Africa were covered with ice. At the great melt-down waters came rushing down the Nile valley, wiping out all forms of life. A hiatus in radiocarbon dates show that the valley was uninhabited for nearly four thousand years, before the first Badarian people moved back in about 6,000 BC. (Some believe the Badarians came from Mesopotamia.) The African ice continued to melt through the millennia and has now almost entirely disappeared.
Abundant evidence shows that the Sahara was once a lush pasture land that gradually dried up. As Flinders Petrie noted in The Pyramids and Temples of Gizeh, Field and Tuer, London, 1883:
The country has undoubtedly been gradually drying up. The prodigious water-worn ravines in the cliffs of the Nile valley show this; and there are remarkable evidences of the Nile having been habitually some 50 feet above its present level, thus filling up the whole valley at all times of the year.
If these meteorological changes affected the Nile valley, how much of Mesopotamia was also affected by these world-wide changes? Did the population also exist in a relatively lush climate that gradually dried up, leading to the development of canals and irrigation systems? Or did human populations move into an already barren land? Isn't that the usual notion of the Tigris and Euphrates river system? Contrary evidence is multitudinous from everywhere around our planet.
If we are to place Old Babylonian mathematics into a broader historical context should we not get ourselves educated on these important world events?
Interpretation of the development of ancient mathematics, based on our modern understanding, led Robson to make several remarks concerning the non-uniqueness of this particular clay tablet. In her arguments about a more commonplace view she stated:
But, although it may be argued that the (re)construction of history is nothing more than inventing more or less plausible stories about the past, each of which will differ according to the historians who tell them, the mathematical artifacts of the past most certainly do not themselves resemble the self-contained settings of a country house mystery. Mathematics is, and always has been, written by real people, within particular mathematical cultures which are themselves the products of the society in which those writers of mathematics live. It is the aim of this article to show how dramatically more convincing a story one can tell about Plimpton 322 if it is put into its mathematico-historical setting.
One of the enduring attractions of Plimpton 322 for the mathematical community has been that it exhibits sophisticated and systematic mathematical techniques for an apparently" pure" end, either "number-theoretical" or "trigonometric."' But a mathematical culture comprises more than its most spectacular discoveries. It is both pernicious and simple-minded to cherry-pick the "cleverest" or "most sophisticated" mathematical procedures (of any society) to present as the history of mathematics. David Pingree has argued that:
the . . . attitude that what is valuable in the past is what we have in the present . . . makes historians become treasure hunters seeking pearls in the dung heap without any concern for where the oysters live and how they manufacture gems. [Pingree 1992, 562].
This is exactly what the purveyors of the wonders of Plimpton 322 have, by and large, been doing hitherto. They have also unwittingly perpetuated the colonization, appropriation, and domestication of the pre-Islamic Middle East by the Western present, as described by Zainab Bahrani:
It is at once the earliest phase of a universal history of mankind in which man makes the giant step from savagery to civilization, and it is an example of the unchanging nature of Oriental cultures. In the Orientalist view the Mesopotamian past is the place of world culture's first infantile steps: first writing, laws, architecture and all the other firsts that are quoted in every student handbook and in all the popular accounts of Mesopotamia. [Bahrani 1998, 162].
Is Plimpton 322 a pearl wrenched from the dung heap of ancient Mesopotamia? Did man make a giant step from savagery to civilization in one sudden leap? Or do we look upon the past from our relatively recent particular notions of human evolution, to thus impose on those ancient people an inverted anachronism that truly fails them? Is Robson herself guilty of taking those ancient civilizations out of a broader historical context because she rejects evidence that disturbs her complacent understanding?
Still another element has affected our view of the history of mankind. We do not recognize multiple cultural influences that came out of the more ancient past, and how the Mesopotamian branches of human endeavor were conditioned from so very long ago. Man did, indeed, come out of savagery. But he was biologically and intellectually influenced by another component that contributed to his path to civilization. This element remains unknown to virtually all of modern scholarship. It had much to do with the origin of the mystery of Plimpton 322.
By no means do I intend to revert to interpretation of the past through modern algebraic notions. But I shall show that Robson's well-constructed paper reaches a truly unwarranted philosophical position. A reinterpretation of Neugebauer's work on Plimpton 322 under a different cloak of symbolism merely suffocates the larger implications of that ancient work. Mathematics is true, no matter how we may symbolize it. Thank goodness. It is an operation of the human mind that has not altered significantly in 4,000 years. It is not subject to the whims of recent scholarship, regardless of how much they may think they are redefining the past. Mathematical procedures may change with time, and human social evolution may follow those changes, but human intellectual operations are still subject to the logic of the apparatus that created them. We did not suddenly blossom into intellectual superiority four thousand years ago; we were merely recovering from a social shock that destabilized all of mankind.
My other references are:
Robson, Hoyrup, and others offer a thorough list of literary references.
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