Marcus du Sautoy explains how mathematical proofs are like narratives, with plots, thrills and ‘whodunnit’ reveals
Mathematicians are storytellers. Our characters are numbers and geometries. Our narratives are the proofs we create about these characters. Many people believe that doing maths is a question of documenting all the true statements about numbers and geometry – the irrationality of the square root of two, the formula for the volume of the sphere, a list of the finite simple groups. According to one of my mathematical heroes, Henri Poincaré, doing maths is something very different: “To create consists precisely in not making useless combinations. Creation is discernment, choice. …The sterile combinations do not even present themselves to the mind of the creator.”
Mathematics, just like literature, is about making choices. What then are the criteria for a piece of mathematics making it into the journals that occupy our mathematical library? Why is Fermat’s Last Theorem regarded as one of the great mathematical opuses of the last century while an equally complicated numerical calculation is regarded as mundane and uninteresting. After all, what is so interesting about knowing that an equation like xn+yn=zn has no whole number solutions when n>2.
What I want to propose is that it is the nature of the proof of this Theorem that elevates this true statement about numbers to the status of something deserving its place in the pantheon of mathematics. And that the quality of a good proof is one that has many things in common with act of great storytelling. My conjecture, if I was to put it into a mathematical equation, is that:
proof = narrative
A proof is like the mathematician’s travelogue. Fermat gazed out of his mathematical window and spotted this mathematical peak in the distance, the statement that his equations do not have whole number solutions. The challenge for subsequent generations of mathematicians was to find a pathway leading from the familiar territory that mathematicians had already navigated to this foreign new land. Like the story of Frodo’s adventures in Tolkien’s Lord of the Rings, a proof is a description of the journey from the Shire to Mordor.
Within the boundaries of the familiar land of the Shire are the axioms of mathematics, the self-evident truths about numbers, together with those propositions that have already been proved. This is the setting for the beginning of the quest. The journey from this home territory is bound by the rules of mathematical deduction, like the legitimate moves of a chess piece, prescribing the steps you are permitted to take through this world. At times you arrive at what looks like an impasse and need to take that characteristic lateral step, moving sideways or even backwards to find a way around. Sometimes you need to wait for new mathematical characters like imaginary numbers or the calculus to be created so you can continue your journey.
The proof is the story of the trek and the map charting the coordinates of that journey: The mathematician’s log.
A successful proof is like a set of signposts that allow all subsequent mathematicians to make the same journey. Readers of the proof will experience the same exciting realisation as its author that this path allows them to reach the distant peak. Very often a proof will not seek to dot every i and cross every t, just as a story does not present every detail of a character’s life. It is a description of the journey and not necessarily the re-enactment of every step. The arguments that mathematicians provide as proofs are designed to create a rush in the mind of the reader. The mathematician GH Hardy described the arguments we give as “gas, rhetorical flourishes designed to affect the psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils”.
The joy of reading and creating mathematics comes from the exciting “aha!” moment we experience when all the strands seem to come together to resolve the mathematical mystery. It is like the moment of harmonic resolution in a piece of music or the revelation of whodunnit in a murder mystery.
The element of surprise is an important quality of exciting mathematics. Here is mathematician Michael Atiyah talking about the qualities of mathematics that he enjoys:
“I like to be surprised. The argument that follows a standard path, with few new features, is dull and unexciting. I like the unexpected, a new point of view, a link with other areas, a twist in the tail.”
When I am creating a new piece of mathematics the choices I will make will be motivated by the desire to take my audience on an interesting mathematical journey full of twists and turns and surprises. I want to tease an audience with the challenge of why two seemingly unconnected mathematical characters should have anything to do with each other. And then as the proof unfolds there is a gradual realisation or sudden moment of recognition that these two ideas are actually one and the same character.
This quality of finding unexpected connections is key to one of the reasons that I love talking about one of my own contributions to the mathematical canon. Some years ago I discovered a new symmetrical object whose contours have hidden in them the complexities of solutions to elliptic curves, one of the great unsolved problems of mathematics. The proof I weave during a seminar or in the journal article I wrote shows how to connect these two very disparate areas of the mathematical world. … Discovering new symmetrical objects in itself is not so difficult. I can get my computer endlessly to churn out new examples of symmetrical objects never contemplated before. The art of the mathematician is to select the ones that tell a surprising story. This is the role of the mathematician, as Poincaré said: to make choices.