How mathematicians can learn from, and even embrace, their mistakes
It’s always nice to be right—but sometimes, says Professor Günter Ziegler, it’s more interesting to be wrong.
“Math starts with a mistake,” says Ziegler, who researches discrete geometry at the Free University of Berlin, and who served as the Master of Ceremonies at the opening of the 4th annual Heidelberg Laureate Forum. Take, for example, Proposition 1 of Euclid’s Elements, a construction of an equilateral triangle that many consider the foundation of modern mathematics. “The solution that Euclid gives to that has been criticized a lot. In modern terms, we could say it’s incomplete or incorrect.”
The fact that Euclid’s work was improved upon doesn’t make his work less valuable. Instead, it shows us the value of preliminary work, on which future mathematicians can build. “Euclid was nevertheless great, and put down foundations of mathematics.”
A few times in his career, Ziegler has learned a great deal from mistakes. Once, a colleague from Scotland wrote to him to point out some apparent inconsistencies in a paper about regular maps. Ziegler’s team scoured the paper, and eventually started to look more closely at a well-respected paper from the 1980s.
“It turned out that the mistake was in the stuff we quoted,” he said. Mathematical researchers build on the foundations of others—but once in a while, new findings rest atop flawed foundations.
The challenge, Ziegler says, is that politeness and even politics can prevent a frank discussion of mathematical errors. The standard referee process—and the publication of errata, which document mistakes—certainly help. Informal online platforms, like arXiv.org, have also expanded discussions of new mathematical research.
But none of these steps are fool-proof—and, Ziegler says, “the interesting part is where we fool ourselves.”
“We spread this image of mathematics, which is: there’s a problem, there’s a solution, there’s a proof, we check it, and then it’s true forever,” he continues. “If you look at modern, high-level research, plainly it’s not so simple.”