Can a picture be a proof? A graph theorist on diagrams, doubt and Lakatos

Most of my working day is spent drawing dots and lines. Graph theory, the mathematics of networks, began with a picture: Euler's paper of 1736 on whether one could walk across each of the seven bridges of Königsberg exactly once. Euler's answer did not depend on the picture, and that is the first lesson every student of the subject learns. The second lesson takes longer. You cannot do graph theory without drawing, and you cannot trust what you draw.

In my exercise classes this produces a question I hear every term, usually from the best student in the room: "If I can see that it's true in the picture, why isn't the picture the proof?" This essay is my attempt at an honest answer.

The case against pictures

The official answer, the one most of us were trained in, is that pictures are aids to finding things out and nothing more. It has a long history. Euclid's very first proposition constructs an equilateral triangle by drawing two circles and taking the point where they cross. Nothing in his postulates guarantees that the circles do cross; the diagram supplies it. In the nineteenth century mathematicians began to find gaps like this everywhere in Euclid. Moritz Pasch, in 1882, wrote axioms for facts about order, such as which point lies between which, that had previously been read off the figure. In 1899 David Hilbert's Foundations of Geometry rebuilt the subject so that no step depends on seeing. He is reported to have said that one should be able to replace "points, lines and planes" by "tables, chairs and beer mugs" without the proofs changing.

Graph theory has its own cautionary tales. In 1879 Alfred Kempe published a proof of the four-colour theorem, built from clever arguments about chains of alternately coloured regions that are easiest to follow in a drawing. It was accepted for eleven years, until Percy Heawood found a configuration in which Kempe's chains behave in a way the pictures had not suggested. Or take the Jordan curve theorem: a closed curve in the plane that does not cross itself has an inside and an outside. Draw one and you will see it is obviously true. Proving it for every such curve, including ones too wild to draw, turned out to be hard, and for a long time Jordan's own proof was regarded as incomplete.

So the case against pictures is not snobbery. Pictures show you a case, and theorems are about all cases. The danger is not that the picture is false. It is that the picture is particular, and quietly lends its particular features to the argument.

Lakatos and the monsters

Imre Lakatos turned this danger into a philosophy. In Proofs and Refutations, a classroom dialogue that has done more for my teaching than any methods course, a class examines Euler's formula for polyhedra: vertices minus edges plus faces equals two. The teacher gives Cauchy's proof, which is essentially a thought experiment with a picture. Imagine the polyhedron made of rubber, remove one face, stretch the rest flat on a table, and count.

Then the counterexamples arrive. A cube with a cubic hole bored through it, like a picture frame, does not satisfy the formula. Neither do some stranger solids the students invent. The class responds in several ways. Some call the counterexamples "monsters" and redefine "polyhedron" to exclude them. Some restrict the theorem. The best students go back to the proof and ask which step failed for the picture frame, and find that the stretching step assumed something the picture made invisible: that the solid has no tunnels.

What I take from Lakatos is that the picture was never the problem. The problem was not knowing which features of the picture the argument used. Once the students knew that, the picture became precise. It told them exactly which solids the theorem covers.

The case for pictures

There is a whole literature of "proofs without words", collected by Roger Nelsen and others, and some of them seem to me to be proofs in the full sense. Draw a single dot, then wrap three dots round it in an L shape, then five more in a bigger L, and so on. You see at once that the first n odd numbers add up to a square, n times n. The picture shows the case of four or five, but you can also see that nothing in it depends on the number you stopped at. Each new L adds the next odd number and makes the next square. The generality is visible.

Berkeley made a similar point about geometry in the introduction to his Principles. When we prove something about a triangle, we draw one particular triangle, with particular sides. The proof holds for all triangles because nothing in it uses the particular lengths. The particular figure stands for the rest, as long as we are careful about which of its features we lean on. Philosophers of mathematics such as James Robert Brown have argued, on similar grounds, that some pictures do not merely suggest theorems but establish them.

I find this convincing, with one condition, which is also Lakatos's. A picture proves when you can say which of its features are doing the work and which are accidents of the drawing, and when you have actually looked for the monster: the case where an accidental feature turns out to have been essential.

What I tell my students now

So my answer to the best student in the room is not "pictures are never proofs". It is a set of questions. What is the picture a picture of: this graph, or every graph like it? Which features did you use? What would the drawing look like if the graph were twice as large, or had a vertex of degree one, or was drawn on a doughnut rather than a sheet of paper? If the argument survives every such redrawing, you may have a proof that happens to be visible. If it doesn't, you have a conjecture with a nice illustration.

The four-colour theorem is now known, through a proof that no human can check by hand and that has since been formally verified by computer. Members here have argued about whether that counts. It is a strange ending for a problem that began with crayons. But I think it proves the point of this essay in a roundabout way. The pictures could not settle it, because nobody could see which features were essential. When the essentials were finally pinned down, there were nearly two thousand of them.

Draw first. Then attack your drawing. What survives the attack is mathematics.