Hello, and welcome to the Thin End of the Wedge. The podcast where experts from around the world share new and interesting stories about life in the ancient Middle East. My name is Jon. Each episode I talk to friends and colleagues, and get them to explain their work in a way we can all understand.
Today’s topic is traditionally the stuff of nightmares for schoolchildren: maths and history together. It’s also enough to make most of us shuffle nervously on our chair. But don’t worry. Today’s guest is an expert in the history of science. He guides us gently through the wonderful world of Mesopotamian maths, and explains its human dimensions.
We look at maths in history. What was mathematics like in ancient Iraq? How did it work? And what did people use it for? And we look at the history of maths. How has our understanding of it changed? And what does the future hold?
People often ask about connections. Was Mesopotamian mathematical knowledge related to that of other cultures? And the big one: can we trace the origins of any modern habits to Mesopotamia? These are difficult questions. They need nuanced answers, and an expert guide.
So get yourself a cup of tea. Make yourself comfortable. And let’s meet today’s guest.
Hello, and welcome to Thin End of the Wedge. Thank you for joining us.
Well, thank you for having me.
Can you tell us please: who are you, and what do you do?
My name is Carlos Goncalves. I study the history of cuneiform mathematics, that is, the mathematics of ancient Mesopotamia. And I work in Sao Paulo, Brazil, at the University of Sao Paulo, where I teach mainly history and sociology of science.
What does Mesopotamian maths look like? What kinds of maths did they have?
The most realistic answer to this question is: Mesopotamian mathematics looks like Mesopotamian mathematics. But I admit this is not really informative. To begin with, we should establish what we mean with the expression “Mesopotamian mathematics”. In general, we use it to refer to a reasonably large set of techniques and strategies. The Mesopotamians used it to count, measure, and calculate. They applied these techniques in professional settings. For instance, in the building of a wall, the digging of a canal, the establishment of workloads. But they also used it in school environment, in hypothetical, more not-so-realistic situations–as, by the way, we do in our school training. This is what I mean with Mesopotamian mathematics: counting, measuring, and calculating.
One of the features of Mesopotamia mathematics that is striking for every one of us that went through high school algebra is the absence of Xs and Ys to represent an unknown number or magnitude. So, for instance, if something is a potential mathematical problem, deals with an unknown length, this length of a field or plot of land rectangle is referred to throughout the text of the problem as “length” without anything like X for length. And this leads us to another way of putting it. Mesopotamian mathematics didn’t use abstract symbolism. Everything was expressed in the language scribes spoke, or in the language scribes knew: Sumerian or Akkadian. The second feature I find really interesting in Mesopotamian mathematics is its relationship between texts and diagrams. I think that for many people, mathematics is always accompanied by illustrations, diagrams, figures. In a large amount of texts from Mesopotamia that refer to geometrical figures or to concrete things such as a wall, a canal, a building, we find no illustrations. This doesn’t mean that there weren’t illustrations that are invisible to the mathematics. It’s only that they didn’t really occur where we expect and this may be due to the fact that we are so used to thinking in terms of our own mathematic tradition, that when we see something different, we find it strange.
Is the idea that the diagrams were drawn in the dirt rather than on the clay tablets?
Yeah, this is a possible hypothesis. And people that do research on cuneiform mathematics have discussed that. The problem is that dirt has not survived to our days. So we cannot be really sure if they did it, if they did it systematically, and how they did it. But it’s a possibility. Sure.
What’s the oldest mathematical text we have? Or the oldest evidence for maths? And when and where does it come from?
This is a nice question. I always feel a little bit insecure of speaking of the oldest object of its kind, because someone else might know of an even older object. Anyway, to my knowledge, the oldest mathematical text that we have from Mesopotamian comes from the ancient city of Uruk. The text dates to sometime between the 34th and the 33rd century before the Common Era; more precisely, to a period known in the archaeological reconstruction as Uruk IV, that spans from 3350 to 3200 BC. Most likely, the text comes from the beginning of this period, that’s to say 3350 BC. We’re speaking of a text that was produced almost 5400 years ago. Just to have an idea of how spectacular the age of this most ancient mathematical text is, let us compare it to another mathematical text from a different tradition that is very well known. It is Euclid’s Elements, from the Greek tradition that was produced around the third century before our era. So if you imagine a line representing time joining the mathematical Mesopotamian text, beginning at 5400 years ago, and joining this point of time to us, Euclid’s Elements would be approximately in the middle of this line. As a matter of fact, it will be closer to us than to the mathematical Mesopotamian tablet.
It is accepted that these first tablets containing mathematical reasonings appeared in the context of the development of the first big cities and the development of bookkeeping. So it seems that all of these appeared almost together and in a way that they related to each other. This is another thing that I find interesting. What’s this most ancient text contain? The text is written on both sides of an approximately rectangular piece of clay that we call a clay tablet. So we have two faces, the obverse and the reverse. And on each side, we can read four measurements that may be lengths and widths of a four-sided plot of land, four-sided field. That’s all that is written. But what is most interesting is that if you make the calculations using these measurement values, and using the techniques Mesopotamians employed for that, you find out that both fields have the same surface, even though the length of their sides are different. One may in this way, suppose that the exercise was conceived to show to an apprentice scribe that fields that have the same area do not necessarily have the same measurements as their sides. So this is surely one of the most ancient mathematical texts from Mesopotamia.
So when you have two problems with the same answer, it’s more likely that this situation was crafted by a teacher than a product of daily life?
Yeah, you are absolutely certain. It’s a strong indication that some mathematical teaching was going on at that time. And if mathematical teaching was going on, at that time, we can assume that mathematics had some social relevance, and possibly, it was used for land measuring. And according to the testimony of other texts, it was used in administrative tasks, such as the delivery of food rations, the control of the amount of grain that they had in silos. So, yes, this is an, so to say, artificial situation. But it seems that taking into account all the other evidence, the teaching of mathematics was just one aspect when they mentioned of the mathematical practices in Mesopotamia at that early time.
Is Mesopotamian maths connected to traditions from other places, such as India or Greece?
Yeah, this is really a thorny issue. I don’t know the answer for certain. I’m no specialist in Indian or Greek mathematical traditions. But to my knowledge, there is no systematic testimony in these traditions explicitly saying they owe some mathematical content to the Mesopotamian tradition. So if we accept as proof of transmission of mathematical techniques from Mesopotamia to India or Greece only a written testimony, then the answer tends to be “no”.
However, there is another way of approaching the problem. It is, in principle, possible that beyond written mathematics, there was some oral, possibly hardly systematised tradition of mathematical practices and possibly mathematical problem solving. This might be valid for Mesopotamia, as well as for India and Greece. If this was the case, and note that this is a huge if, then the road is open to play with possible transmissions.
Which then would be the contents that are the best candidates to having been the result of transmission? One answer to this question is what we call second degree equations. A second degree equation is a kind of equation. This name, second degree equation, is not anything that we find in ancient texts. And it’s equation that has a variable. Usually in textbooks, this variable is called X. And this variable appears with an exponent 2, the square of X. The procedure to obtaining the solution or the solutions to this type of equations, proceeds through very similar steps, producing very similar intermediary results when we compare what we have in the Mesopotamian tradition, and some other traditions, especially the Arabic tradition; not really exactly the same thing, as we see in the Greek tradition.
Yet, it must be clear that this is a hypothetical. The fact that two different cultures were able to solve what we call second degree equations in structurally equivalent ways is no proof that they communicated. To illustrate this point with another example, [a] more trivial example: consider the situation: if two students that sat a mathematical examination solve a problem in the same way, this is no proof that they cheated. So the problem of exploring oral traditions is that we have no option than to study the possible imprint orality made on written texts, except by reading written texts. And it’s often very dependent on personal opinion, if a characteristic of a written text comes from an oral tradition or not. Anyway, one interesting last thing to take into account is that none of these mathematical traditions were monolithic. In the case of Mesopotamia, if we compare the mathematics of the Old Babylonian period and the mathematics of the Seleucid period, we find differences. So when searching for evidence, even if you find evidence of transmission between cultures, we must be clear on what part of the tradition we’re looking at. And in my personal opinion, skepticism must prevail.
How about counting in 60s? It’s commonly said that the modern habit of measuring time in 60s and having 360 degrees in a circle comes from Mesopotamia. Isn’t that true?
Yeah, it might be true, or it might at least be partially true. Because of the possibility of transmission of this way of writing numbers and writing measurements of time that might have occurred during Seleucid times when the last centuries of the Mesopotamian tradition got in touch with the Greek tradition. So it might be true. It’s a possibility that these 60s come from–this way of representing numbers representing time–comes from Mesopotamia. But we cannot say from this that Mesopotamians in their daily life, used to measure time as we do with our clocks and watches. So we have to nuance this statement in a certain way.
How do Mesopotamians write numbers?
This is a lovely question, but maybe before answering it, I could explain to the non-specialists how Mesopotamians wrote in a few words. The writing tools of Mesopotamians were a stick possibly made of wood, reed or of some other organic material, and a piece of clay, round or square, that was in most cases flattened into a somewhat pillow shape. We call such a piece of clay “a clay tablet”. In order to write, when pressed one end of the stick against the surface of the tablet. Contrary to the way we write nowadays with a pencil, or pen, Mesopotamians didn’t drag their writing tool on the surface of the writing medium. They pressed the stick on the clay leaving very tiny marks with an approximate triangular shape. So the first modern scholars that tried to understand the writing of Mesopotamians used the Latin word to describe this triangular shape–the Latin noun cuneus that can be more precisely translated as “a wedge”. So, this is why we call their writing cuneiform. So, this is the basic shape used to write in the Mesopotamian writing. This shape, the wedge, were made on the surface of the clay in various directions. Also, there are in fact two different types of wedges: a wedge properly speaking, with a thick end and a theme end; and a different wedge from which thin end was lacking. In other words, a wedge that [had] only the thick end and no thin end.
And now I can say how Mesopotamians wrote numbers. They used normal wedges in a vertical position, thick end up, thin end down, to represent units. And the other type of wedge, the thick end only wedge, to represent tens. For instance, in order to represent the number 37, they inscribed on the surface of the clay three thick end only wedges to represent 30, followed by seven vertical regular wedges, somewhat stacked one onto the others, to represent seven, thus obtaining 37. So this is the basic operation, they used to write numbers.
Did Mesopotamians have numbers as we would understand them?
As a matter of fact, we understand numbers in several senses. Philosophy of mathematics has many different positions as to what a number is. So I will restrain myself to the way of writing numbers. So, Mesopotamians wrote numbers in a way that is not exactly what we do, but can be related to what we do. There was something special in their system. But before entering into the details, I’d like to invite the listeners to reflect on the way we place the positions, we place the different digits of our own system of writing numbers. When we have 10 units, we start writing tens at the left of the units. When we have 10 tens, we start writing the hundreds at the left of the tens, when we have 10 hundreds, we start writing thousands at the left of the hundreds and so on. So every time we reach 10 of something, we open a new writing position to the left, so that we can write something 10 times bigger. That’s why we speak of decimal. Decimal comes from 10. It’s Latin. That’s why we speak of decimal places. And our mathematics teacher at high school insisted that we write in a decimal base or base 10.
So Mesopotamians did something similar, but they use sixties instead of tens. So we speak of a sexagesimal system. When Mesopotamians reached 60 units, they started writing 60s to the left of the units. When they reached 60 sixties, they started writing 60 times bigger numbers to the left of the sixties. And every time they reached 60 in the writing position, they started writing in a different newer place, immediately to the left of the previous place. The Mesopotamians used what our mathematics calls sexagesimal place value notation. A very good comparison is the way we write minutes and seconds. Every time we have 60 seconds, we add one minute. This is very similar to the way Mesopotamians wrote numbers. This is the basic system for writing numbers.
One mistake that is often seen in school textbooks that try to teach something about ancient mathematics is the idea that this was the only way Mesopotamians wrote numbers. On the contrary, more precisely, when Old Babylonians counted things like portions of the year, sheep and a batch of clothes, they used different arrangements of wedges and thick end only wedges to represent specific numbers like 10, 60, 60, 3600, 36000. And there was a different system to represent surface measurements, and also many other systems and system combinations that we see in the other periods of Mesopotamian history.
It sounds potentially quite confusing to have lots of different counting systems. We might wonder how they knew which one to use.
Yeah, apparently, they did know, it’s really confusing for those of us that do research on these most ancient periods. There are lots of different systems, lots of different notations sometimes for the same thing. So the impression of confusion is legitimate, but probably people that used those systems, or at least some of those systems, I believe they were more at ease doing that. It’s a problem of historical perspective, let us say.
But what about the difference between those simple sexagesimal numbers you mentioned versus the different systems for measuring areas or whatever else?
Yes, you are right, especially in the Old Babylonian period. The Old Babylonian period is a period of the history of Mesopotamia that roughly begins at 2000 BC, and finishes in 1600 BC. It’s the period of Hammurabi. This is the period I am more acquainted with. This is the period by the way where most of the mathematical tablets that we know come from. There seems to have existed the specialisation of this way of writing numbers with the sexagesimal place value notation, and the other systems use it for counting and measuring, especially in the tablets from Nippur. But this is not really exactly the same in other areas.
The sexagesimal place value notation is really more frequently used when scribes had to make some calculations, had to specifically multiply numbers. So it’s really common to have a text with the measurements given–measurements of a field, for instance, or a geometric figure–so the measurements are given in one of those different systems. Then the scribe converted these measurements into a sexagesimal place value notation, made the calculations with the sexagesimal place value notation, and sometimes converted the result back to one of those specialised systems. In the region of the Diyala that I studied, this is not always so. We find examples of multiplications, where the scribe used the measurement values. So, it is as if the scribe multiplied one metre by one metre, obtaining the value directly using the measurements. And of course, they didn’t use the unit of measurement metre. It’s just an example to try to make it clearer. So there was this tendency to use the sexagesimal numbers to make multiplications, and to use the other systems, the other notations, to present data and to present the final results.
How has our understanding of Mesopotamian maths changed over the years?
Yeah, so our understanding of Mesopotamia mathematics comes from the activity of researchers that devote all or most of their time, in order to read, understand, analyse, and interpret the mathematical texts that came down to us from Mesopotamia. That understanding is shaped by personal interests, by different epistemological positions, and by the institutional settings as well as the historical settings of each of us, researchers, historians of mathematics, or assyriologists with an interest in the history of Mesopotamian mathematics.
So in very general lines, I like to understand the history of this research in the following way: around the 1940s and 1950s, the emphasis was on understanding what kind of structures Mesopotamian mathematics had. This is the moment when researchers used to say that Mesopotamian scribes were able to solve second degree equations, even though scribes didn’t call what they were doing second degree equations. The 1990s are very important in the history of how our understanding of Mesopotamian mathematics evolved. During these years, the researchers understood a lot of details about the emergence of the most ancient mathematical practices, mainly in the city of Uruk, some 5000 years ago, exemplified, by the way, by that very ancient tablet I mentioned before. More precisely, the interest of researchers led them to associate bookkeeping practices with the emergence of many mathematical notations and tools.
Also, in the 1990s, we saw a lot of attention given to the specific vocabulary of Mesopotamian cuneiform texts. This is the moment where understanding was made that Mesopotamian mathematics had conceptually different types of addition. For instance, one addition were one operation to add something to an existing and changeable entity, much in the way we add some money to our bank account and our bank account continues to be our bank account. Another addition where Mesopotamians add symmetrical things, so to say merged, as when one adds the contents of two silos of grain in order to know how much grain is at one’s disposal. This concern grew, this study of the language of Mesopotamian mathematics was one of the reasons why some concern grew, that saying Mesopotamians knew how to solve second degree equations was a little bit misleading. Researchers were worried that the Mesopotamians’ specificities of the concepts of addition and multiplication, that the absence of abstract algebra and notation, all of this could not be described simply by resorting to an object, so load[ed] of meaning as our present day, second degree equation. So people started to find alternatives, such as problems on squares and rectangles. Another preoccupation of that period was whether the emphasis in our interpretation of these Mesopotamian mathematical problems should fall in the algebraic aspect, the geometric aspect, or the algorithmic aspect. Those were very rich years for the research on the history of Mesopotamian mathematics.
Reaching the recent years, I find a situation much more difficult to analyse, because researchers in the field have been extremely active, and many new publications are scheduled to appear in the following years. I discern at least three emerging interests among researchers. First, in what extent was Mesopotamian mathematics constitutive of the social life. And secondly, were Mesopotamian mathematical practices different in different social settings? The mathematics of the school is different from the mathematical and administrative section in the palace, in the temple. And so how was this distributed over society? And finally, was there a connection? And how strong was this connection between these different social settings, especially between the mathematics that we see in scribal schools, and the mathematics of professional scribes working in those organisations such as the palace and the temple, where there were, so to say, departments or sections or at least officials dedicated to store and distribute commodities and other goods?
What are you working on at the moment?
At the moment I’m working on this first issue. Was mathematics constitutive of the social life? And how can we approach this problem? I’m working on texts that are not mathematical in the strict sense. They are loan contracts of barley and silver, mostly. But they have lots of numbers. And they have lots of people involved in those transactions. And I’m trying to relate those two dimensions: the numerical, the mathematical, the measurement knowledge, and the way people associated among themselves. I’m using a little bit of social networking analysis, but not only this. I still have students working on the history of mathematics in the strict sense and the more traditional sense here in Sao Paulo.
How can we follow your work?
People interested in my work are welcome to follow my new page at the academia.edu website. And if the interest is more specific, I will be so glad to answer emails.
Thank you very much.
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