When people describe a story as being told in a "by the numbers" fashion, that's usually regarded as bad. But in a talk at Oxford University the other day, scientist Marcus du Sautoy argued that a great mathematical proof is a lot like a story. In fact, a really great proof is like Frodo's journey in Lord of the Rings, from the familiar to the new.
Du Sautoy explains:
"A proof is like the mathematician's travelogue. Fermat gazed out of the 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 the mathematician has already navigated to this foreign new land.
[It's] a bit like Frodo's adventures in 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 then bound by the rules of mathematical deduction, like the legitimate moves of a chess piece, prescribing the steps you're permitted to take through this world.
At times, you'll 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 around for new mathematical characters — like imaginary numbers, or the calculus — to be created, so you can take that next step. The proof is the story of the trek, and the map charting the coordinates of that journey. The mathematician's log."
You can watch the whole event, which was livestreamed and recorded for posterity, over at Oxford University. Besides du Sautoy's 20-minute talk, there are responses by author Ben Okri, mathematician Roger Penrose, and literary scholar Laura Marcus, and an audience question-and-answer session. [Oxford University]