I.
The article is “The two cultures of mathematics in ancient Greece,” and it’s by Markus Asper, a contemporary historian. Let’s read it, as it highlights some tensions that are worth poking at.
Quick: first thing that comes to your mind when I say “ancient Greek math”!
What did you come up with? Was it,
- Euclid?
- proof?
- geometry?
- Pythagoras?
- irrational numbers?
If you thought of any of these words, congratulations! You’ve located the theoretical branch of Greek math, which is precisely one of the two cultures that Asper points to in this piece.
In fact, his whole point is that it wasn’t just Euclid, Pythagoras, proof, etc.:
The words of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance…Recently, however, a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.
The rest of that iceberg wasn’t theoretical math. Everyone else was doing practical math, and much of this article reviews the careful historical clues we have as to what practical math looked like in ancient Greece.
The fact is that we do need clues, because the theoretical camp were the elites. They wrote the books, they dominated our picture of mathematics today. And the theoretical mathematicians that we know and love today had no love for practical mathematics. The mathematics of Euclid was an elite activity that was at a comfortable distance from anything that might be mistaken as professionally useful. Practical mathematics — algorithmic, procedural, professional, useful — was seen as common and not proper.
Nowadays you can enter any educational conference and hear a thousand sessions with something like the following message: “Don’t teach people to just follow procedures! Teach them to prove, to generalize, don’t just tell them how to do it — help them understand it!”
Euclid would have likely agreed, but Euclid wasn’t the only mathematician hanging around Athens. There was a whole collective of lower-class mathematics engaged in practical math, and they were all about following procedures.
II.
Who were these practical mathematicians?
For one, they were pebble calculators. In ancient Greece, pebbles were used to perform difficult calculations (see Netz) and there were likely professional pebble calculators, but we don’t know much about them — like most ancient professionals, they rarely made it into literature. There are hints here and there, and also archaeological evidence of counters, but it’s not much.
(Netz makes the very cool point that coins as a technology are essentially a riff on the counter. A coin has the manipulative properties of counters, but they’re also made of precious heavy metals that are imbued with intrinsic value.)
Beyond numerical calculators in ancient Greece, there were also geometric measurers and calculators — experts in surveying area and volume.
If you have read anything about ancient Mesopotamian mathematics, what the Greeks did will feel familiar. As it should, because their methods of measuring and calculating area and volume were almost certainly an inheritance from their Near East neighbors.
In other words, ancient Greek math included things like this:
Concerning stones and things needed to build a house, you will measure the volume according to the rules of the geometer as follows: the stone has 5 feet everywhere. Make 5 x 5! It is 25. That is the area of the surface. Make this 5 times concerning the height. It is 125. The stone will have so many feet and is called a cube.
These texts have several features. First, they sound like someone is trying to teach you something — that’s because they are! Second, they sound mired in specifics, and it might be tough to learn general procedures all on your own. That’s also right. This was valuable knowledge, so it was protected. You’d need a teacher to help, an apprentice, to help you really get the hang of things. Asper explains:
Since this kind of knowledge was of economic importance, it was probably not popular or widespread but rather guarded, perhaps by guild-like social structures. Performing as a practical mathematician in one of these arts was a specialized profession. For some of these people, a Greek name has survived: there was a professional group called harpedonaptai (‘rope stretchers’), obviously surveyors operating with ropes for measuring purposes.
I don’t mean to blur distinctions between the past and present, but I do think it’s worth taking a moment to think of all the professions today that might be called practitioners of practical mathematics. We’d be looking for people who aren’t hunters of knowledge, proof, or generalization — we’re looking for people who are calculating, measuring, counting.
Computers have changed things, obviously, but they haven’t done so entirely. I’d say actuaries and accountants probably ought to make the cut. Probably not cashiers, in this day and age. Actual surveyors, definitely. Computer programmers? Arguably.
How is math used in other professions, especially those that don’t require a college degree? I don’t really know. Probably not much more than measuring a cup of this, 5 ml of that, counting 9:30 plus 3 hours. I’m just guessing here — I’d like to know more about practical math used in common professions.
III.
And what about theoretical mathematics? It’s what you’d think. I’ll quote some juicy bits:
Perhaps it would be adequate to think of theoretical mathematics as some form of game rather than something pertaining to a professional occupation, which it as become today, and which practical mathematics has always been. The persons who played this game were certainly at home in the upper circles of Athenian society (evidence collected by Netz 1999a, 279f), similarly to Plato and his followers who eagerly absorbed theoretical mathematics. From the majority’s perspective, comedians could already make fun of mathematicians in 414 BC. They must have felt like an elitist little group among Athenians. For them, theoretical mathematics was probably a status practice, perhaps enforced by the fact that the most common status practice, that is, politics, became quite dangerous for the old upper class at the end of the fifth century. Mathematics was, as philosophy was to become, a status-conscious way to keep one’s head down.
Here is Aristophanes making fun of Meton, an astronomer/geometer:
PISTHETAERUS In the name of the gods, who are you?
METON Who am I? Meton, known throughout Greece and at Colonus.
PISTHETAERUS What are these things?
METON Tools for measuring the air. In truth, the spaces in the air have precisely the form of a furnace. With this bent ruler I draw a line from top to bottom; from one of its points I describe a circle with the compass. Do you understand?
PISTHETAERUS Not the very least.
METON With the straight ruler I set to work to inscribe a square within this circle; in its centre will be the market-place, into which all the straight streets will lead, converging to this centre like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.
…
PISTHETAERUS I want to give you a proof of my friendship. Use your legs.
METON Why, what have I to fear?
PISTHETAERUS ‘Tis the same here as in Sparta. Strangers are driven away, and blows rain down as thick as hail.
METON Is there sedition in your city?
PISTHETAERUS No, certainly not.
METON What’s wrong then?
PISTHETAERUS We are agreed to sweep all quacks and impostors far from our borders.
METON Then I’m off.
What makes theoretical mathematics distinctive — the proofs, the generality, the passive-aggressive distancing from anything practical — to Asper is a tell:
I suggest that many of the odd features of the theorists, such as expressly refusing to mention any practical applications or any useful effect, worked intentionally as distinctive markers, meant to distinguish the precious game of distinction from sordid occupations that were carried out by people for hire. Plato himself defines, quite polemically, the difference between practical calculations and theoretical mathematics…A late and, almost certainly inauthentic anecdote illustrates my point nicely:
“Someone who had taken up geometry with Euclid, asked after he had understood the first theorem: ‘What is my profit now that I have learned that?’ And Euclid called for his servant and said: ‘Give him a triobolon, since he must always make a profit out of what he learns.”
Theoretical mathematics, the playground of the elite.
IV.
Here is the final tally of the distinctive features of practical and theoretical mathematics in ancient Greece:
Greek practical math:
- was derivative of older traditions that, ultimately, originated in the ancient Near East;
- solved ‘real-life’ problems;
- communicated actual procedures in order to convey general methods;
- used written texts (if at all) as secondary means of knowledge storage and instruction;
- employed ‘social’ technologies of trust, that is a rhetoric based on institutional authority; for example, the guild’s pristine tradition, the specialist status of its practitioners, and the knowledge’s commonly accepted usefulness;
- worked within a stable and highly traditional social–that is, institutional–framework;
Greek theoretical mathematics:
- emerged in sixth- to fifth-century Greece, at least partly from a practical background
- was a theoreticians’ game with artistic implications, pointedly removed from ‘real life’;
- communicated general theorems concerning ideal geometrical entities;
- depending on writing and produced autonomous texts;
- employed epistemological technologies of trust based on evidence and logic;
- was not institutionalized, at least no during its formative stages.
So with all this, what exactly are we doing in schools today? What are students learning?
It’s a weird mix, which I suppose reflects the weirdo development that led us here. If you were in a schoolhouse in the 19th century in the US, my sense is that you were mostly studying calculation with a practical eye. You’d learn some basics and then solve problems with an eye towards applications in business or the market. Usually, that was that.
If you were wealthy, you went to secondary school or even university. And there you were certainly studying theoretical mathematics — even ancient Greek theoretical mathematics. But then in the 20th century, we greatly expanded secondary school enrollment. So now there’s all this theoretical math that everyone is studying.
Though the way we instruct all these kids is more in line with the way practical math was taught (“used written texts as secondary means of knowledge storage and instruction”). And now even the practical stuff is not practical. And to bridge the gap between high school and elementary school, we have moved more of the theoretical work into ever earlier grades. Not that this is a bad thing, mind you.
Overall, the distinctions of ancient Greek mathematics don’t map cleanly on to school math today, I don’t think.
V.
I had two purposes in this blog post. The first was to just share and summarize a really interesting article. Go ahead, use it however you’d like.
The second purpose was to help me put my finger on a tension that I keep on seeing.
If I got up in front of a bunch of people in math education in 2019 and told them, “too many people only think of a few famous men when they think of Greek math,” there would be agreeable nodding.
If I then continued and said, “in fact, people in every culture and class were practicing mathematics in ancient Greece in ways that history, with its focus on elites, tends to ignore,” I’d get some some sympathetic social media shares.
If I went further and said, “we really need to respect the many different cultures of mathematics, and not pretend that math always looks the same in every culture,” that’s something I could get some applause with, if I could borrow some keynoter charisma, I think.
And all of those things are true. But if I then said, “many mathematical cultures don’t even care at all about why something is true, and they also don’t care about proofs or making theoretical generalizations — they just want to know how to solve the problem,” well well well right this way Mr. Pershan, yes very interesting alright have a nice day we’ll call you.
And I think that tension is generally present when we correctly expand our notions of who does math and what math looks like. Looking at the ancient Greek example makes me think that we really ought to respect practical mathematics — which by definition is mathematics that is not concerned with the “why.”
And yet there is so often disdain among some teachers for “mindless” calculation or “thoughtless” problem-solving. That seems unfair to me.
There are any number of reasons why we might want to teach students the “why” or proof or generalizations — the theoretical stuff that is woven into our teaching, and is perhaps the profession’s dominant value. But I think it’s wise to ask ourselves if we can respect our students’ own mathematics — even if it’s practical and disinterested in proof or justification, and even as we then make moves to help them see a bit more theoretically. Because theoretical mathematics was born of elitism.
? What. if. we. just. made. one. up.
and 
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Teaching With Problems 