In ancient Mesopotamia and Egypt, mathematics meant law and order. Specialised mathematical technocrats were deployed to settle conflicts regarding taxes, trade contracts, and inheritance. Mathematics enabled states to develop civil branches of government instead of relying on force and violence. Mathematics enabled complex economies in which people could count on technically competent administration and an objective justice system.
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How did geometry start? Who was doing it, and why, in early civilisations? The Greeks invented theorem and proof, but long before them there was geometry in Egypt and Mesopotamia. So that’s practical geometry, applied geometry.
Or is it? Actually even the oldest sources have lots of pseudo-applications in them. Such as: Find the sides of a rectangular field if you know the perimeter and the diagonal. Or: I have two fields, and I know how much grain each field produces per unit area, and I know the total grain produced by both of them, and I know the difference between their areas, now tell me how to find the area of each field.
Not the kind of situations you find yourself in every day exactly. You can judge for yourself if that deserves to be called applied mathematics. Given obscure and convoluted information, find something that should have been much easier to measure directly than this artificial data you somehow had access to.
In any case, geometry like that, whatever you want to call it, was highly developed almost four thousand years ago. Why? What made people do this? Let’s try to find out.
Early mathematics emerged where there was fertile soil. Rivers that made this possible. Agricultural abundance meant resources enough to expend some people specialising in mathematics instead of having all hands on the ploughs.
Look at a modern population density map of Egypt. You will find that virtually the entire population is concentrated along the Nile; all the rest is pretty much desert. That’s still today. Even with the assistance of modern technologies the river area is by far the most liveable. Even more so back then when geometry started, thousands of years ago.
It was the same in Mesopotamia, present-day Iraq. Also a river civilisation with very good agricultural conditions. They had legendary gardens that were praised in ancient sources. Google it: The Hanging Gardens of Babylon. You will see some nice pictures of what these luxurious gardens might have looked like. That’s a nice visual for this idea that it was agricultural abundance that made a specialised pursuit like mathematics possible in those societies.
So that explains why they had the resources to support mathematics. But why would they want to? What did they stand to gain from geometry?
Basically, mathematics was for a long time about commerce and taxes; bureaucratic management of workers and produce; inheritance law. Those kinds of things.
Eleanor Robson’s book is very illuminating about this. “Mathematics in Ancient Iraq: A Social History”, the book is called. She emphasises especially that mathematics was very strongly associated with justice. A society without a functioning justice system is hampered by constant disputes about land, taxes, inheritance. Everybody is fighting with everybody. Like the old American West, you board yourself up and mind your own business and if there’s a disagreement, well, that’s what guns are for, isn’t it?
Mathematics is the way out of this primitive state. Mathematics is objective. It can settle these disputes in a fair way. If everybody is wasting a huge amount of effort and resources on petty disputes in a lawless no-man’s land, who you gonna call? The mathematicians, that’s who. That’s how it went in ancient Iraq.
A specialised, highly trained mathematician would come in and delineate all the plots of land, compute all the taxes owed, and distribute every inheritance. All according to exact calculations. This stuff used to be ruled by emotions, personal animosity, and the law of the jungle. But now, thanks to mathematics, that is replaced by objective rules. Who can argue with a calculation? Mathematics takes the worst sides of human nature out of the equation.
When society is run by fair, universal rules, people no longer have to constantly look over their shoulder and fear that some lawless eruption of force could destroy everything they have at any moment. A functioning justice system enables people to work for the collective good and to plan for the long term.
It is the authority of mathematics that makes this possible. These skilled mathematical technocrats had great credibility because people recognised that they were above the subjective and the emotional. They were bound by dispassionate calculation. Mathematics compelled them to be fair and rational.
Indeed they explicitly said so themselves. As one mathematical scribe put it: “When I go to divide a plot, I can divide it; So that when wronged men have a quarrel I soothe their hearts. Brother will be at peace with brother.” That’s a quote by one of those mathematical technocrats, explaining what geometry accomplishes. Note that it has both of those elements I emphasised. Mathematics is the opposite of emotional disputes. It soothes heated hearts, it creates peace between warring brothers. And the quote also highlights that this happens because of the expertise of the mathematician: I know how to do this kind of thing, the technocrat is saying. It takes special training.
The quote is from Eleanor Robson’s book. Here’s another thing she points out that is yet more evidence of the importance of mathematics in this context. The Sumerian word for justice literally means straightness, equality, squareness. Also in Akkadian: justice is the “means of making straight.”
Again, another major indicator of this: “the royal regalia of justice were the measuring rod and rope.” Think of those Lady Justice statues that you see sometimes. She’s blindfolded because that shows that she’s unbiased, and she has these scales, showing that she’s considering both sides and weighing them carefully and fairly. That’s the symbol of justice in our society. But, in ancient Babylon, the symbols of justice were not a blindfold and a set of scales. Instead, Lady Justice was a geometer. She held her land-measuring tools. Those were the instruments of justice in ancient society.
Maybe it’s pretty much the same today, four thousand years later. Back then, the trustworthiness of mathematics was a cornerstone of society. If people didn’t trust mathematics, there could be no law and order, no state bureaucracy, no complex economy, no civilisation. Today, that link is perhaps less evident. But perhaps no less crucial. We have added many layers of complexity to our society, but perhaps looking back at historical societies is the same thing as looking into the inner essence of our own. Maybe without faith in mathematics the entire fabric of our society would unravel. Maybe without mathematicians mediating their disputes, “brother would be at war with brother” as that ancient scribe feared.
It is interesting also that this role of mathematics that I have outlined is really as much psychological as it is scientific. What makes this whole system work is not only that mathematics can give useful answers to certain technical problems. The psychological side is equally essential: mathematics has a kind of aura of objectivity, of trustworthiness, of professional expertise. That goes well beyond merely calculating the taxation rate of some field, or how many goats you can buy for a silver shekel. The system rests on a more nebulous trust in the mathematician class by the population at large. The idea of mathematics, the image of mathematics, is more important than the sum of its actual applications.
That’s an important conclusion because it explains that striking feature of ancient mathematics: namely that many of the problems the ancients texts solve are super fake. They are pseudo-applications.
For instance: Find the two sides of a rectangle, given that the sum of the length and the width is 24, and that the area plus the length minus the width = 120. So in other words, you basically have two equations in x and y, and if you solve for y in one and plug it into the other you have a quadratic equation in x. Lots and lots and lots of problems like that in Babylonian mathematics.
Obviously nobody would ever face a problem like that in any real-word situation. It’s very often like this: you are looking for something simple, like the sides of a rectangle x and y, and you are given some super weird, like some convoluted combination of x and y is three eights of some other convoluted combination of x and y.
Here’s another actual one: The width of a rectangle is a quarter less than the length. The diagonal is 40. What are the length and the width?
In what real-world scenario can you realistically end up knowing the diagonal of a rectangle, and the difference between the sides, but not the sides themselves? And why couldn’t you just measure the sides? Someone did measure the diagonal, apparently, so why not the sides?
Sometimes these texts hardly even try to hide how fake they are. One problem goes: I found a stone, but did not weigh it. I cut away one-seventh and then one-thirteenth, and then it weighed so-and-so much. What was the original weight of the stone?
Who among us has not “found” whatever random stone, then chipped away an extremely exact ratio of it, and then suffered some kind of stone-cutter’s remorse I guess, and tried to reconstruct the original weight of the stone for some reason.
Very relatable, isn’t it? Actually it kind of is. Not because we are sitting around cutting one-thirteenth out of random stones, or because we are running around measuring the diagonals of various fields and then later wish we had measured the sides instead. That never happens to any sane person in the real world. But it does happen in math books. Still today, we torture our students with such questions, one more artificial and unrealistic than the other.
Some people think that kind of thing is modern pedagogy run amok. They see these kinds of problems in modern textbooks and they think: How silly modern pedagogy has become! These naive educators are bending over backwards to make math “relevant” to kids, but the just end up with silly fake problems.
History offers a different perspective. The problems may be silly, but the cause is not a misguided obsession with real-world relevance among modern educators. Fake problems are as old as written mathematics itself. For as long as there has been mathematics education, students have been forced to go through page after page after page of pseudo-problems that only superficially, or linguistically, appear to be talking about real-world things, while actually corresponding to absurd scenarios that would never happen.
In a way one might argue that history vindicates these problems. They are not so silly after all, if we consider them in the light of the role of mathematics in ancient Babylonian society. Mathematics doesn’t support the economy merely by keeping the account books. It’s more than that. Mathematics is what instills confidence in monetary law and order, without which any kind of complex economy would be impossible in the first place.
For this system to work, there needs to be a specialised class of number-crunching technocrats. These people need to embody logic and reason and objectivity. They need to be math machines, detached from politics and emotion. A long schooling in artificial pseudo-problems makes some sense as a means of creating this class.
From this point of view, it is even a strength that these problems are artificially divorced from real-world problems, because the mathematical technocrat is supposed to be detached from such concerns anyway. Mathematicians are valuable to society precisely because they are so disinterested in the needs of people of flesh and blood. It is this disinterestedness that makes people willing to trust the mathematicians to be the arbiters of disputes.
The sheer volume of training in pointless problems also has its point. It is not enough that people at large know some mathematics: they could use mathematics as a tool for evil, as just one more incidental weapon in a society still ruled by greed and conflict. For a complex economy to take off, there needs to be faith that the law and the state administrative bureaucracy are fair and consistent. This faith comes from the credibility of mathematics. The mathematical technocrats need to be proper experts to justify the confidence placed in them. They need to embody mathematics; they need to single-mindedly look at any situation or conflict and see only the mathematics in it.
Society needs the mathematicians to not only get the right answer, but to have great authority as proper experts. And it needs them to be “nerds,” so to speak, who are so one-sidedly developed that they can only see mathematics anywhere they look, and not let emotions or politics influence their work. A long and rigorous training in fake applied problems is not a bad recipe for bringing this about. Arguably, we pretty much still use the same recipe to the same end today, thousands of years later.
So that’s the Babylonian tradition. We know quite a bit about it because they wrote on clay which is pretty durable. In Egypt, mathematics was recorded on papyrus, which isn’t going to survive for thousands of years normally. So we only have two or three or maybe four papyri that beat the odds and were conserved. But it seems the Egyptian situation may very well have been quite similar to the Mesopotamian one in terms of the role of the mathematicians.
“Geometry” means “earth-measurement.” That’s from the Greek: geo metria. The ancient Egyptians had the same idea but their word for it was more concrete: a geometer was literally a “rope-stretcher.” A land surveyor stretches ropes to measure distances and delineate fields.
A rope is pretty much equivalent to a ruler and compass. Pull the ends of the rope and you have a straight line. Hold one end fixed and move the other one while keeping the rope stretched: now you have a circle.
Euclid explains how to make a square with ruler and compass. That’s Proposition 46 of the Elements. The Egyptians would have done that long before with their stretched ropes. Try it for yourself, it’s fun: go out into a field with a friend and try to make a perfect square using nothing but a piece of string. You will see why geometers were called rope-stretchers.
Do you think you could make a square? Do you think anyone could? Back in the day, this skill could have given you a leg up in life. Suppose you make one square field, and then a rectangular field with the same perimeter. The square field will have greater area. But you could trick those less knowledgeable in mathematics. You could say: you get that field and I get this one, fair and square. Just try it for yourself, you would say, let’s walk around the fields and count the number of steps. 400 steps around my field, 400 steps around yours: aha, our fields are the same size. That’s what you tell the other guy, who isn’t such a math person. But you know that of course 100*100 is way more that 50*150. So later you get a much greater harvest. But of course you would pretend that that’s because you worked so hard while the other guy was lazy. Maybe that’s another way in which ancient society is like ours: privileged people use their privilege to rig the game in their favour, and then pretend it was all due to merit.
According to Proclus, this kind of mathematical deceit did indeed happened: “The participants in a division of land have sometimes misled their partners. Having acquired a lot with a longer periphery, they later exchanged it for lands with a shorter boundary and so, while getting more than their fellow colonists, have gained a reputation for superior honesty.”
Here’s how Thomas Heath paraphrases this in his History of Greek Mathematics: “Proclus mentions certain members of communistic societies who cheated their fellow members by giving them land of greater perimeter but less area than the plots which they took themselves, so that, while they got a reputation for greater honesty, they in fact took more than their share of the produce.”
A dubious paraphrase, in my opinion. Can you spot the suspicious part of it? Good old Heath put something in there that was not in the original source. Hint: turn to the title page of Heath’s book. There are some clues there. The book was published by Oxford University Press in 1921. Heath’s name comes with some bells and whistles: it’s Sir Thomas Heath, in fact, and then K.C.B, K.C.V.O. That’s Knight Commander of the Royal Victorian Order etc.
Titles upon titles. It’s an establishment guy, this Sir Thomas. A gentleman scholar, who was a civil servant as his day job at the Treasury.
What part of Sir Thomas’s paraphrase of the ancient mathematical land deceit reflects his own social context more than that of the ancients he is trying to describe? I’m thinking of his phrase that these were “communistic societies.” The original source says nothing at all about this having anything to do with communism. But you can understand how Sir Thomas would have been concerned about communism at this time. The Russian Revolution started in 1917, Heath’s book is published in 1921. While writing the book, Heath was a secretary at the British Treasury. He would have read all about Lenin and Bolsheviks in The Times while having his afternoon tea. And those worries would have been at the top of his mind when he sat down in his study to do his scholarly work in the evening. It didn’t take much provocation, one imagines, for him to have a swing at how “communistic societies” were dreadful and corrupt.
We must always read historical sources this way. Context matters.
Now, the “original” in this case was Proclus. But that’s not much of an “original” to speak of. Proclus is nobody. He’s not particularly trustworthy. He was writing in the year 450 or so, thousands of years after the historical events he is talking about. So it’s anybody’s guess how much truth there is in what he is saying. And in any case, like so many other mediocre writers, both ancient and modern, Proclus is just copying what others had said.
Let’s illustrate this point. Let’s see what we can learn by looking at Proclus’s account of the origins of geometry in Egypt. Here’s what Proclus says:
“Geometry was first discovered by the Egyptians and originated in the remeasuring of their lands. This was necessary for them because the Nile overflows and obliterates the boundary lines between their properties. It is not surprising that the discovery of this and the other sciences had its origin in necessity, since everything in the world of generation proceeds from imperfection to perfection. Thus they would naturally pass from sense-perception to calculation and from calculation to reason. Just as among the Phoenicians the necessities of trade gave the impetus to the accurate study of number, so also among the Egyptians the invention of geometry came about from the cause mentioned.”
Ok, sounds pretty plausible. But it’s worth running Proclus through a plagiarism checker, just as we do with modern student essays these days. Cutting-and-pasting from Wikipedia is nothing new. Proclus had many Wikipedia equivalents available to him. Perhaps he stole the whole thing for example from the Geography of Strabo, which was written more than 400 years before. Here’s what Strabo says:
“An exact and minute division of the country was required by the frequent confusion of boundaries occasioned at the time of the rise of the Nile, which takes away, adds, and alters the various shapes of the bounds, and obliterates other marks by which the property of one person is distinguished from that of another. It was consequently necessary to measure the land repeatedly. Hence it is said geometry originated here, as the art of keeping accounts and arithmetic originated with the Phoenicians, in consequence of their commerce.”
Basically a dead ringer for the Proclus passage. Plagiarism detected, SafeAssign™ would say.
Actually Proclus has added something that is not in Strabo, namely the claim that this historical episode illustrates how human though passes from the world of the senses to the higher realm of reason. This is card-carrying Platonism. Proclus is a sycophantic follower of Plato. He sees everything through Plato-coloured glasses. Which is not helpful if we want to use him as a source of historical information. As Heath had his anti-communism, so Proclus has his Platonic axe to grind and it infects everything he says.
Actually we can go back even earlier than Strabo. Let’s take an equal jump back in time again: another 450 years still. From Roman Strabo to classical Greek Herodotus. He too speaks of the origins of geometry in Egypt. Let’s listen to his account:
“This king [Sesostris] also (they said) divided the country among all the Egyptians by giving each an equal parcel of land, and made this his source of revenue, assessing the payment of a yearly tax. And any man who was robbed by the river of part of his land could come to Sesostris and declare what had happened; then the king would send men to look into it and calculate the part by which the land was diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, in my opinion, the Greeks learned the art of measuring land.”
Ok, I have to admit that this makes Thomas Heath look a bit better. “The king gave to each an equal parcel of land”: That is a bit more like communism. Heath said he was paraphrasing Proclus where there is no such phrase about equality. But Herodotus, the better source, kind of vindicates him a bit. You could imagine, in the scenario that Herodotus describes, that certain administrators in charge of implementing the king’s decree might secure a nice big square plot for themselves and trick the mathematically illiterate into a smaller plot with the perimeter trick. Perhaps not entirely unlike how corrupt middle-managers in the Soviet bureaucracy might manipulate the system for personal gain. But be that as it may.
I think there’s another interesting thing about Herodotus’s description compared to Strabo’s. Strabo and Proclus give a cleaner and simpler account: the flooding of the Nile obliterates everything and you have to start afresh each year with the drawing of boundaries. Herodotus’s account is much less dramatic: some parts of properties might become damaged by the floods, and the task of the mathematician is not to redraw the entire agricultural map each year but rather to calculate what proportion of area has been lost in each case for taxation purposes.
One can easily imagine how a desire to simplify and tell a clear and dramatic story might have led authors like Strabo and Proclus to prefer their version. The older source is a bit more “boring” but perhaps that makes it more credible.
Indeed, Herodotus’s account fits better with what we said about the role of mathematics in Mesopotamian society. In Herodotus’s version, the mathematicians task is more technical, more specialised, more bureaucratic. Note his phrase: “the king would send men” to do the calculations. You have to send mathematicians. They are a small, specialised class of technocratic experts that are dispatched to solve disputes with authority and objectivity. That’s precisely the main point I have made today, so let us end there.