Reading a math paper is a bit like having dinner at a nice restaurant. The entrée might taste delicious, but it doesn’t tell the full story of how it was made. Clever recipes that end up tasting funky don’t make the menu; undercooked dishes are (usually) not served to customers. But missteps in both conception and execution are important parts of the process. Danny Calegari, a topologist at the University of Chicago, wants mathematicians to be more transparent about what goes on in the kitchen.

Calegari grew up in Melbourne in a mathematical household. His father taught calculus and his mother statistics at a local technical college. Their bookshelves were filled with textbooks that he and his little brother Frank would browse through for fun.

But Calegari didn’t always plan to be a mathematician. When he enrolled as an undergraduate at the University of Melbourne, he considered becoming a writer, or perhaps a cognitive scientist. He decided to take a topology class to better understand some models of how the brain worked. “It was just electrifying,” he said. “In this class, every sentence was somehow interesting. … It was like the world’s greatest Lego set or something, where it all just connected up in this brilliant, unexpected, powerful way.”

He was hooked. He decided to major in math, and later got his doctorate from the University of California, Berkeley, ultimately sticking with topology. (Frank would follow him to Berkeley and then join him on the math faculty at Chicago, specializing in number theory.)

Calegari remains an avid reader — of Joyce, Dickens, Nabokov, Austen, Vonnegut, Lorrie Moore, Raymond Queneau, Anne Carson. He also writes short stories of his own from time to time. But mostly, he’s drawn to math, to problems in low-dimensional topology and geometric group theory that keep him awake at night. “Once I had this encounter with topology, which was a little bit like getting run over by a train, everything changed,” he said.

In a recent issue of the Notices of the American Mathematical Society, Calegari published an essay about the importance of failure in mathematics. Disappointment, he wrote, “is both a crisis and an opportunity.”

Quanta spoke with Calegari about what failure in mathematics looks like, and why it’s important. The interview has been condensed and edited for clarity.