We tend to think of STEM as the very essence of modernity. The technology that STEM makes possible has created modern life and is shaping the future. But fundamentally many aspects of STEM, how it defines our understanding of the world and relates to society, go back thousands of years. This blog entry explores the earliest days of STEM and how it compares to today by looking at one of the oldest extant mathematical texts, the Rhind Papyrus.
The Rhind Papyrus was found in Thebes, Egypt and was probably written around 1650 BCE. It is an ~17 foot long papyrus scroll written in hieratic, a less formal writing system than hieroglyphics. It contains two tables of fractional equivalents followed by 84 worked problems. The problems vary from moderately abstract calculations involving linear algebra and basic geometry to very applied problems. The extensive applications provide an intriguing look at both Egyptian mathematics and the problems and organization of Egyptian life.
Consider the first table of fractions. For every odd integer from 5 to 101, it gives a decomposition of twice the unit fraction (1/n) into a series of distinct unit fractions. For example, 2/5 is decomposed into 1/3 + 1/15. Throughout the papyrus, similar unit fractions are used. This suggests that unit fractions were viewed as customary, stable quantities that provided a basis for further calculation. In some sense, only integers and unit fractions were seen as real numbers. This illustrates something fundamental but often overlooked, that mathematics is not merely a set of procedures but an imaginative act. It is the willful decision to seek out order, quantify and segment the world into units that are we find meaningful. Moreover, this process of creating unit systems is not just an ancient discovery. Similar systemization continues. On a societal level, we continue to redefine what units and quantities are preferred and given pragmatic value. The rise of the metric system and current pervasiveness of binary are both creative ways to redefine quantity to suit current systems of understanding and pragmatic needs.
The other thing we see in the fraction table that still resonates today is the sense that mathematics has an aesthetic element. All the fractional equivalents are constructed to follow certain rules. For example, no unit fraction is repeated. This is similar to the requirement in many classrooms that students reduce all fractional answers to the simplest form. Such reduction does not affect the accuracy of the answer, but it does create a sense of completion that is aesthetically pleasing.
Although the Rhind Papyrus shows evidence of the creative and aesthetic elements that make mathematics engaging, in ancient times, as now, math was not an abstract science. It was highly practical. The primary reason the fractional tables are part of the papyrus is to facilitate solving the 84 problems that are the core of the text. These problems cover a large range of topics from dividing bread among a group of people to finding the volume of a pyramid. In looking at the content of the problems, we have an intriguing look at Egyptian society and at the role mathematics played in its functioning.
One of the major uses of mathematics in ancient Egypt was running an economy. Some of the problems show this in quite simple ways, calculating the volume of fields or dividing food rations. These problems show the daily use of math in administering a complex society. But there also indications of a more complex relationship between math and economics. Probably the most entertaining problem in the papyrus is Seven houses each have seven cats. Each cat catches seven mice. Each mouse would have eaten seven ears of corn. Each ear of corn would have produced seven hekat of grain. How many things are mentioned all together. This is a somewhat silly problem but it is also significant because it suggests one of the uses of math was to quantify future yields and assess losses. Moreover, doing so involved multiple factors including those at a substantial remove. It is, in a sense, among the earliest forerunners of the type of economic forecasting that has become a major area of applied mathematics.
The other major area we can clearly see math being applied is in engineering. When most people think of ancient Egypt, they first envision their major engineering achievements – the great temples and pyramids. In the geometry problems from the Rhind Papyrus, we see the basic calculations that allowed such projects to be planned.
Then, in the contents of the Rhind Papyrus we see the rudiments of both the abstract, creative and applied sides of mathematics. But we can also learn something from the role this document played in its own time. This papyrus probably served as a training manual and reference for a scribe. It is easy to think about scribes as merely recording the words and events of others, but they were actually the technocrats of the ancient world. Scribes were the educated class who had the literacy and numeracy to be able to enact the will of those in power. So, at the time it was created, this scroll would likely have fulfilled a role similar to a modern technical handbook. As one historian put it:
A 17-foot roll like the Rhind Mathematical Papyrus would have cost two copper deben, about the same as a small goat. So this is an object for the well-off. But why would you spend so much money on a book of mathematical puzzles? Is this the Ancient Egyptian version of our craze for Sudoku? The answer is … not quite. Because to own this scroll would, in fact, have been a very good career move. If you wanted to play any serious part in the Egyptian state, you had to be numerate. A society as complex as this needed people who could supervise building works, organise payments, manage food supplies, plan troop movements, compute the flood levels of the Nile – and much, much more. To be a scribe, a member of the civil service of the pharaohs, you had to demonstrate your mathematical competence.
This papyrus then shows us how competence provided the currency of membership in a technical class that was between the laborers and the rulers. It one of the earliest examples of how intellectual capacity could be converted into social position.
To understand a mathematics that is creative and orderly, that provides an aesthetic sense of completion, that is pragmatic and promotes career growth. These ideas are remarkably resonant 3500 years later. These are many of the same things we still wish to provide for our students. But most of all, the core goal of both research and education in STEM is to access what the original Egyptian title purports that the text contains: The Correct Method of Reckoning, for Grasping the Meaning of Things and Knowing Everything – Obscurities and All Secrets.
Leibowitz (2018, June) The Rhind Papyrus. Retrieved from http://www.math.uconn.edu/~leibowitz/math2720s11/RhindPapyrus.html
O’Connor, J.J. & Robertson, E.F. (2000, December) Mathematics in Egyptian Papyri. Retrieved from http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_papyri.html
Toomer, G.J. (1971) Mathematics and Astronomy, in J.R. Harris (ed.), The Legacy of Egypt (pp. 37-40), Oxford: Oxford University Press.
MacGregor, N. (2012) A History of the World in 100 Objects. London: Penguin Books.