When I was in college, I spent a lot of time with a certain science major who just couldn’t seem to get the Zen of mathematics. Whenever I would say something like “there is no formula for the roots of a fifth-degree polynomial”, she would reflexively correct “You mean, there’s no formula *you know about*.”

But no, I really mean that there is *no* formula. I mean it is *impossible* to find such a formula. And when a mathematician doesn’t say “impossible” lightly; it doesn’t mean “no one’s ever done it” or “it’s prohibitively hard” or “it seems hopeless”. It means impossible.

It might seem hard to believe that people can ever know things are impossible. How is mathematical impossibility different from garden-variety closed-mindedness?

But there is more going on here than “I’ve seen a lot of things, and if never seen something like [X], then [X] doesn’t exist.” There’s a lot more. In fact, often one can learn a lot more from proving a problem has *no* solution than from solving a problem.