During his career, the German mathematician Carl Jacobi made important contributions to several scientific fields. In his work, he often solved difficult problems by following his maxim “Man muss immer umkehren,” which loosely translates into “Invert, always invert.”
Jacobi believed that one of the best ways to clarify your thinking is to restate problems in their inverse form. He would write down the opposite of the math problems he was trying to solve and found that the solutions often came more easily to him.
While Jacobi mainly applied inversion to mathematics, it’s an equally powerful approach in other areas. No matter what problems you’re trying to solve, it can help you uncover errors and roadblocks that aren’t readily apparent.
The way to use inversion is to think about things backward instead of forward. Rather than asking how to do something, you ask how not to do it. Let’s have a look at some examples of what those questions might look like.
If you’re like most people, you rarely ask yourself questions like these. For most of us, inversion is highly counterintuitive. Thinking backward is not something that comes naturally to us.
But it’s very much worth learning because it helps improve your understanding of problems. It forces you to step out of your usual way of thinking, consider different perspectives, and come up with new solutions.
While inverting problems won’t always solve them, it will help you avoid trouble. When you know what you don’t want, you can take steps to make sure those things don’t happen. And that will move you closer to the solution.
So, whenever the best path forward isn’t clear — flip the problem on its head. Instead of trying the find the right path to take, make sure you know which ones to avoid.
Keep in mind: “Invert, always invert.”
Spend less time trying to be smart, and more time trying to avoid being stupid. Avoiding mistakes is much easier than seeking excellence — and it’s usually a much better way to solve your problems.