You’d be hard pressed to escape school without having to learn some kind of geometry. In the history of ideas and popular imagination geometry is associated with the ancient Greeks. Long after calculus and algebra have fled the mental nest, the Pythagorean theorem continues to linger. But what if it wasn’t illustrious Greek philosophers but Babylonian bureaucrats who first invented geometry? A new paper, published this month and making waves in the international press, argues not only that a 3,800-year-old clay tablet is familiar with Pythagorean principles, but that it was used in practical settings in the ancient world.
The tablet in question is an unassuming, partly broken clay tablet named Plimpton 322 that is roughly the size of an iPhone. It was donated to Columbia University by educational publisher George Arthur Plimpton, who had acquired it in 1922 from diplomat-cum-antiquarian Edgar James Banks (the inspiration for Indiana Jones) for the unprincely sum of $10. On the basis of its handwriting, distinguished professor Eleanor Robson dates the tablet to around 1800 B.C., making it roughly 4,000 years old. The tablet itself contains a table (14 columns x 15 rows) written in cuneiform (wedge-shaped) script. From our modern perspective it is noteworthy that the table uses what are now called Pythagorean triples (i.e. integers a, b, c that a2 + b2 = c2. The most common example that you probably remember from school is 3, 4, 5).
What this means is that long before ancient Greeks and Indian mathematicians formulated their own solutions, the Babylonians had an understanding of geometry. Mathematician Daniel Mansfield, a senior lecturer at the School of Mathematics and Statistics at UNSW Sydney and author of the headline-grabbing new article, told The Daily Beast that people “knew very simple geometry (areas for rectangles and right triangles) from the very earliest times. Then, during about the Old Babylonian period (1900-1600 BCE) they became interested in ‘Pythagorean triples,’ which are special rectangles and right triangles that have simple measurements.” Plimpton 322 is famous for this technical development but there are several thousand other mathematical texts from ancient Iraq that can demonstrate the development of mathematics in the region.
Though Plimpton 322 and its importance has been well known among scholars of Ancient Near Eastern science and literature for many years, there is some debate about its origins. One big question for modern historians is, who made this tablet? In her prize-winning work on this Robson notes that unlike our own society, “Ancient Mesopotamia was a culture that prized anonymized tradition over individual creativity.” That said, Robson states, we can deduce some details about its creator. He was almost certainly male (the tablet is said to have come from Larsa and most female scribes worked further north). He was not a mathematician in the modern sense as there were no professional mathematicians until “relatively recently” and he does not seem to have been an ancient elite who liked speculating about numbers. Neither phenomenon, Robson says, existed in Mesopotamia’s 3,000-year history: “He must have been someone who used literacy, arithmetic, and mathematical skills in the course of his working life.” As a result of this, Robson suggests that the tablet’s maker was a professional bureaucratic scribe, a teacher, or perhaps both.
Mansfield suggests a slightly more precise and professional context for the tablet’s production. He told me that while the tablet “is a theoretical text” it was “inspired by the problems of the day.” It might have been a schoolroom text, he said “but it would not have been very useful for the role.” The reason for this is that “educational problems were usually created in reverse, starting with a nice simple answer. Plimpton 322…does not contain the characteristic simple answers. Instead, it looks like someone generated all the Pythagorean triples they could, and then studied their sides.”
Mansfield hypothesizes that the tablet was created in the context of ancient surveying practices. Surveyors had to take accurate measurements of the length, area, and right angles, of spaces in order to avoid disputes about land ownership and tenure. To do this they “used the sides of the rectangles (or right-angled triangles) known as Pythagorean triples.” Mansfield told me that contextualizing the tablet in surveying praxis can help explain “why the Mesopotamians were so interested in Pythagorean triples.”
Mansfield’s work is hardly the “discovery” some outlets are calling it. Many of his arguments are rearticulations of the work of Robson and others, and this has prompted some pushback from a scholarly community justifiably frustrated that STEM is taking credit for the work of humanists. Plimpton 322 is one of the most famous ancient mathematical artifacts that we have. What is interesting about theories of its formulation is that they draw attention to the way that scientific, mathematical, and intellectual discoveries often take place in collaborative contexts. Mansfield told me that he envisioned that the tablet was the result of teamwork involving a group of people, at least some of whom would have been professionally trained as scribes. They would have worked together to develop the table and the information it contains. Dr. Jeremiah Coogan, a postdoctoral fellow at Oxford University who works on tables, agrees. He said that work on Plimpton 322 reveals the “contingency” of mathematics. Like Robson, he sees mathematics as historically embedded and contingent working in “practical contexts of bureaucracy, administration, and other large-scale co-ordination…in none of these contexts was mathematics the province of an isolated scholar in their study.”
This is striking because this is precisely the opposite way that we tend to describe scientific discoveries today. We credit Pythagoras with his theorem, Archimedes with the hydrostatic principle (about buoyance and the displacement of water), and Euclid with geometry. (Notice that there are no Babylonians here). But discovery isn’t always (if ever) the product of inspired moments in the bath. As Durham University classicist Serafina Cuomo has explored in her introduction to Ancient Mathematics, the history of mathematics involves practical collaborative work like land-surveying and accounting as well as seemingly mundane tools like tables, abaci, sundials, and sighting instruments. There are often many more people involved than we think: tools like the abacus or the reference table,” Coogan said, “can be physically manufactured by [people other than their uses] and that information can be produced, refined, and distributed by others.” There are a lot of cooks in the kitchen.
The history of mathematics, Cuomo argues, is not just about the lofty mathematical communities that sprung up around important figures. The distinctions we make between pure and applied math that emerge out of antiquity and we now bandy about as description are informed by political, socio-economic, and national divides. Focusing on the mythical great mathematicians isn’t just inaccurate it misrepresents the historical contexts in which mathematics was performed and steals credit from a larger (and sometimes more ancient) group of collaborators.
All of which is to say is that mathematics is both collaborative and practical, which may be bad news for the middle school student who protests that they will never use algebra.
