“Singing Euclid: the Oral Character of Greek Geometry”, 2020-06-21 (; backlinks; similar):
Greek geometry is written in a style adapted to oral teaching. Mathematicians memorised theorems the way bards memorised poems. Several oddities about how Euclid’s Elements is written can be explained this way. Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or the script of a play: it was something the connoisseur was meant to memorise and internalise word for word. Actually we can see this most clearly in purely technical texts, believe it or not. It is the mathematical details of Euclid’s proofs that testify to this cultural practice.
…Here’s an example of this, which I have taken from Reviel Netz’s book The Shaping of Deduction in Greek Mathematics. Consider the equation A + B = C + D. Here’s how the Greeks expressed this in writing: THE
A AND THE B TAKEN TOGETHER ARE EQUAL TO THE C AND THE D. This is written as one single string of all-caps letters. No punctuation, no spacing, no indication of where one word stops and the next one begins. A Greek text is basically a tape recording. It records the sounds being spoken…Modern editions of Euclid’s Elements are full of cross-references. Each step of a proof is justified by a parenthetical reference to a previous theorem or definition or postulate. But that’s inserted by later editors. There is no such thing in the original text. Because it’s a tape recording of a spoken explanation. Referring back to “Theorem 8” is only useful if the audience has a written document in front of them …Consider for example Proposition 4 of Euclid’s Elements…“The triangle will be equal to the triangle”, says Euclid: this is his way of saying that they have equal area. After Euclid has stated this, he goes on to re-state the same thing, but now in terms the diagram…This is exactly the same thing that he just said in words. But now he’s saying it with reference to the diagram. He always does this. He always has these two version of every proposition: the purely verbal one, and the one full of letters referring to the diagram…That’s something of a puzzle in itself, but here’s the real kicker though. Not only does Euclid insist on including the abstruse verbal formulation of every theorem, he actually includes it twice! This is because, at the end of the proof, his last sentence is always “therefore…” and then he literally repeats the entire verbal statement of the theorem. It is literally the exact same statement, word for word, repeated verbatim. You say the exact same thing when you state the proposition and then again when you conclude the proof. Copy-paste. The exact same text just a few paragraphs apart.
…So what was the value of this very expensive business of repeating the statement of the proposition? The oral tradition explains it. The verbal statement of the proposition is like the chorus of a song. It’s the key part, the key message, the most important part to memorise. It is repeated for the same reason the chorus of a song is repeated. It’s the sing-along part. In a written culture you can refer back to propositions and expect the reader to have the text in front of them. Not so in an oral culture. You need to evoke the memory of the proposition to an audience who do not have a text in front of them but who have learned the propositions by heart, word by word, exactly as it was stated, the way you memorise a poem or song. This is why, anytime Euclid uses a particular theorem at a particular point in a proof, he doesn’t says “this follows by Theorem 8” or anything like that. He doesn’t refer to earlier theorems by number or name. Instead he evokes the earlier theorem by mimicking its exact wording. Just as you just have to hear a few words of your favorite chorus and you can immediately fill in the rest. So also the reader, or listener, of a Euclidean proof would immediately recognise certain phrasings as corresponding word for word to particular earlier propositions…In one case it is even irrelevant that the remain sides are equal as well, but Euclid still needlessly remarks on this pointless information in the course of the proof of Proposition 5 even though it has no logical bearing on the proof. Go look up Euclid’s proof if you want to see this nonsense for yourself.
…It’s pretty fascinating, I think, how textual aspects that appear to be purely technical and mathematical, such as a few barely noticeable superfluous bits of information in the proof of Proposition 5, can open a window like this into an entire cultural practice.