The hero lives their life, but twists, turns and complications arise. They meet helpful or harmful characters, in a world governed by its own rules, its own internal logic. In the end, unexpected events occur and the quest moves forward. At last we understand something we did not understand before.

The same is true of a proof. Faced with a familiar situation, we may come up with a few small concrete examples, but then we must generalise and solve a much broader problem, at first as huge as a mountain. Yet thanks to the scrolls of the Ancients, containing magical formulas that work under certain conditions, after some work, the solution appears and we finally understand.

Remembering a proof is very similar to telling a story. You have to identify the key moments, the timely intervention of this or that character, an old bearded Greek and his theorem, applying under such and such conditions, so that the internal rules transform the situation all the way to a happy conclusion. But if one detail is forgotten, the whole story may collapse.

Marking one’s path, leaving traces — breadcrumbs or little stones — whether fragile or lasting, is something many tales and mathematics have in common. It allows us to find our way back, if we can rely on them. Otherwise, we risk getting lost.

Taking notes is crucial in mathematics. We write out the steps of the calculations. This helps relieve working memory, freeing up space to process other information. If there is a mistake, we can go back and check where we missed something. If there is success, following the same path will lead to another success, more quickly the second time.

The strategies adopted by the hero are varied. Their task is rarely achieved on the first try. They try again and again, from a different angle each time, showing flexibility in their approach. This transforms the hero: they gain experience, like the three little pigs, whom we may see as three versions of the same character maturing with each attempt. Failure transforms you and can improve you. Solving a substantial problem requires perseverance, originality, flexibility and effort.

This is true in tales and in mathematics. A good story is worth its weight in sweat. And as Corneille wrote: “To conquer without danger is to triumph without glory.”

Being quick or slow, big or small, is neither good nor bad when facing a problem. What matters is taking time to think, postponing an obvious decision, and showing critical judgement. Because what seems simple is often more twisted, in tales. One must prove one’s worth, just as a mathematical statement requires a rigorous, formal proof. A mere accumulation of evidence is not enough. Lies may hide behind what seems obvious, a small problem may turn out to be hard to solve, and a major problem may be solved by a clever idea, simple but powerful.

The world of a tale may not be obvious at first. We have to assume things, accept certain axioms: animals can speak, characters can be as tiny as a thumb, and this creates a context in which certain tricks will make sense according to the internal logic of the story.

The same is true of most mathematical notions: we must accept that things work this way or that way. Sometimes, “I don’t understand” simply means “I do not accept your axioms.” We have to realise that sometimes there is nothing to understand, only a few axioms to accept as they are; otherwise the story cannot unfold.