The use of diagrams in geometry raise questions about the place of the physical, the sensory, the human in mathematical reasoning. Multiple sources of evidence speak to how these dilemmas were tackled in antiquity: the linguistics of diagram construction, the state of drawings in the oldest extant manuscripts, commentaries of philosophers, and implicit assumptions in mathematical proofs.

Transcript

Diagrams. What are their role in geometry? Some people like to think that the logic of a geometrical proof doesn’t need the diagram. Mathematics is supposed to be pure and absolute. Diagrams seem connected to the visual, the intuitive, that makes it kind of psychological, and perhaps therefore even subjective.

Certain people don’t like that association one bit, so they try to minimise the role of diagrams. Maybe diagrams are just crutches to help those with weaker minds, whereas a perfect logical reader could follow the proof from the text alone. Some people like to think so. It’s a dogma that fits modern tastes.

But, historically, that interpretation is a pretty poor fit. In some ways, classical geometry appears to have embraced visuality rather than tried to replace it with abstract logic.

There are signs of this attitude in the very language of Greek geometry. The word for proving is the same as the word for constructing: grafein, to draw. To prove something is literally to make it graphic. And a theorem, in ancient Greek, is a diagramma, a diagram. Instead of the Pythagorean Theorem the Greeks would say the Pythagorean Diagram.

Indeed there is always one diagram for each theorem in Greek mathematics. That’s a very rigid rule. In modern mathematics we often find it natural to have several pictures for some proofs, and no pictures at all for many other proofs. Just do what comes natural to explain the particular content. But not the Greeks. One theorem, one picture: this rule was extremely firmly ingrained in their conception of geometry.

And not only in geometry, in fact. Euclid follows this rule slavishly even when he writes about number theory. For example, he proves (Elements VII.30) that if a prime number divides a times b, then it divides either a or b. A very important theorem that is still proved in every modern book on number theory. But no modern book would include a picture for this. It’s just not a visual thing at all, so it makes little sense to draw a picture to go with it.

But Euclid does. The numbers that he is talking about he draws as line segments. The bigger the number the longer the line. But this has little to do with his proof. The proof is not visual. It’s just as abstract as the ones in the modern books. So the diagram doesn’t really do anything. And it’s like that theorem after theorem after theorem: Euclid has these useless diagrams that are basically irrelevant to the content. But he insists on the rule “one theorem, one diagram” even where it doesn’t really seem to serve any purpose.

At least it doesn’t serve any purpose in terms of capturing or visualizing the steps of the proof. Maybe it has other purposes. One purpose could be to signal that number theory is subsumed by geometry. The number 5 really just means a line segment of length 5 units, Euclid seems to be saying with these diagrams. So since Euclid has established the foundations of geometry, and number theory so to speak lives within geometry, then it follows that Euclid has established the foundations for number theory as well. Number theory doesn’t need separate foundations since it is subsumed by geometry. Maybe this is what Euclid is trying to emphasize with his pictures of numbers.

Or maybe Euclid needs pictures because he doesn’t have algebra. A modern proof of theorems like these are very dependent on algebraic notation. If p divides ab, then p divides a or b. In the course of the proof you keep referring to relationships between these number all the time. Suppose p divides ab but not a. Etc., etc. It would be hard to get all that across without algebraic symbols.

If you have a picture you don’t need algebra, because you can point. Instead of the letters a, b, p you have line segments of different lengths that you can point to and say: suppose that one dives that one. You don’t need algebraic symbols or letters, because you are pointing to a picture. The mode of presentation is oral; you have your audience in front of you, and you have drawn the diagram in the sand with a stick, and you point to it as you reason your way through the proof.

You might say: But Euclid does have labels, like A, B, C, etc. So he is referring to entities by letter or label designation, not merely by pointing visually. Well, maybe. But one could argue that that’s not really what Euclid’s A, B, Cs mean.

When Euclid calls things alpha, beta, gamma, it is perhaps inaccurate to translate this as A, B, C. Because it would also mean 1, 2, 3, or first, second, third. The Greeks wrote numbers this way, using the letters of the alphabet. Alpha meant 1, beta meant 2, and so on. So perhaps we shouldn’t think of Euclid’s ABC as algebraic designations. Perhaps it simply means “the first point,” “the second point,” and so on.

This makes it seem a lot closer to the pointing hypothesis. Perhaps the standard way for mathematicians to explain their reasoning was to point to a picture and say “this one,” “that one,” and so on. Then to encode this in writing they used alpha, beta, gamma, to mean “the first one I mentioned,” “the second one I mentioned,” and so on.

If this is right, then the letters in the English version of the Elements are a bit deceptive. They seem more algebraic, more modern, than they really are. From that point of view, diagrams in number theory make some sense.

In fact, in early modern geometry, in the 17th century, you sometimes see people labeling points in diagrams 1, 2, 3 instead of A, B, C. Because they thought this was the right way to translate Greek into Latin. Euclid’s alpha is really a 1, and so on. They were more sensitive to Greek culture back then. Nowadays people have forgotten about that stuff.

Here’s another fun linguistic-cultural perspective on diagrams in Greek geometry. The language in which Euclid describes constructions is quite odd. “Let the circle ABC have been described.” The language of Greek mathematics “makes the author and temporality disappear from a proof,” as one historian has put it. Euclid is not saying that he’s drawing the diagram, and he’s not telling the reader to draw the diagram. He’s just sort of commanding the diagram into existence.

You know the book of Genesis in the Bible: “Let there be light,” God said, and there was light. Euclid uses literally the same kind of construction. It’s exactly the same verb form as in the Ancient Greek version of the Bible. Just as God makes heaven and earth by merely pronouncing that they exist, so Euclid makes geometrical objects appear just ordering them to be. It’s not “I draw” or “you draw” but “let it have been done.”

You could read this as supporting a Platonic conception of mathematics. Euclid is distancing himself from actual drawing. The objects of mathematics just are. They are not something you or I have to make.

But here’s a counter argument to this interpretation. Netz argues that actually Euclid’s grammatical construction merely reflects a purely practical circumstance of the Greek tradition. Namely, that Greek mathematicians had to prepare their diagrams in advance due to technical limitations of the visual media available. Here’s what Nets writes:

“Of the media available to the Greeks none had ease of writing and rewriting. [Standard media were papyri and wax tablets, and, for larger audiences, such as Aristotle’s lectures,] the only practical option was wood painted white. None of these [ways of representing figures] is essentially different from a diagram as it appears in a book. The limitations of the media available suggest the preparation of the diagram prior to the communicative act---a consequence of the inability to erase. This, in fact, is the simple explanation for the use of perfect imperatives [such as] ‘let the point A have been taken’. It reflects nothing more than the fact that, by the time one comes to discuss the diagram, it has already been drawn.”

That’s Netz’s interpretation, and if he’s right then Euclid’s grammatical choice reflects only incidental cultural circumstances and says nothing about philosophical commitments.

So “let it have been done” just means “I did it yesterday”. It doesn’t mean that geometry is set apart from concrete action and that doing has no place in mathematics.

It’s fascinating how the same aspect of the text takes on such a different meaning when cultural context is taken into account, compared to a purely philosophical reading. In fact, let me tell you about another striking aspect of Greek manuscripts which is also like that. Namely, the way diagrams are drawn in manuscripts of Greek geometry.

Diagrams in manuscripts of Greek mathematical treatises are very often very poorly drawn. They are oversimplified and crudely schematic. Ellipses, parabolas, and hyperbolas are represented as pieces of circles and so on. Very poor pictorial accuracy.

Also the simplicity and specificity of the diagrams often obscure important mathematical points. For example, the figure for the Pythagorean Theorem is often drawn in manuscripts with the two legs of the triangle being equal, even though the theorem holds for any right-angle triangle. The diagram thereby gives the misleading impression that the theorem is less general than it really is.

So you might think: aha, clearly the Greeks didn’t care about the diagrams. They are poorly executed, poorly thought through. So diagrams couldn’t have