Back in the 1800s, a German by the name of Carl Jacobi made waves in the world of mathematics. His seminal contributions to elliptic functions, number theory, differential equations and determinants would shape the direction of the field for centuries to come.
Jacobi was known for his approach to identifying research opportunities and is said to have told his students that when seeking a research project “man muss immer umkehren”. For those not familiar with German, this translates roughly to “Invert, always invert.”
What Jacobi meant, of course, was that inverting equations could open up new possibilities in the field of mathematics. But as countless writers have noted, the principle of inversion can be usefully extended as a mental model to other problems.
The Inversion Technique
Inversion is a way of thinking about what we want by considering it in reverse. In other words, while conventional forward thinking would ask what we need to do to get what we want, inversion encourages us to think about what might prevent us from getting what we want.
The principal objective of inversion is to avoid the mistakes that can put achievement at risk. One can be so focused on what is explicitly required to achieve their goals that stupid – and crucially, avoidable – mistakes end up costing them that goal.
Put differently, one can be so focused on driving toward their end destination that they stupidly forget to stop at the red light. Turning a problem upside down can help us avoid driving through that red light.
But importantly, inversion also helps us to see problems from a different perspective. It encourages fresh thinking and ultimately can stimulate innovation. Turning a problem upside down can not only stop us driving through that red light, but it can also help us to identify shortcuts.
Examples of Inversion
To illustrate the point of inversion, let’s look at some simple examples in two of the focus areas of this blog: productivity and personal finance.
Suppose we have a goal to improve our productivity.
With conventional forward thinking, we would ask the obvious question: “What are some things that would improve my productivity?”
Here are a few examples of what we might conclude:
- Use prioritisation tools, e.g. the Eisenhower matrix
- Allocate time for deep work
- Adapt my environment to reinforce positive habits
- Use short-term rewards to maintain motivation
Conversely, inversion provokes the reversed question: “What are some things that would destroy my productivity?”
Examples of what we might conclude here include:
- Spending too much time consuming low-quality information
- Leaving my mobile beside me while working
- Setting vaguely defined goals
- Spending too much time with procrastinators
Of course, the first set of ideas are important, solid suggestions for improving productivity. But inversion captures some other important considerations around our use of technology, our types of goals, and even our relationships. If we don’t avoid these things, we have little chance of improving our productivity.
#2: Personal Finance
Suppose now that we have a goal to improve our financial health (a worthy goal for most).
With conventional forward thinking, again we would ask the obvious question: “What are some things that would improve my financial health?”
Some wise suggestions might include the following:
- Investing into a well-diversified portfolio
- Tracking my money
- Understanding the psychology of money
- Automating some investment decisions
Of course, by now you know where we are heading next. Inversion reverses the question: “What are some things that would destroy my financial health?”
Examples might be:
- Expensive borrowing
- Holding cash only
- Conspicuous consumption
- Panic buying
Again, both sets of ideas are helpful, but if we don’t avoid the things brought out in our inversion exercise, we have scarce chance of improving our financial health.
Using Inversion in Practice
These are simple, common-sense examples to illustrate the point. Most people sensibly considering their financial health and productivity will surely have considered some of the ideas generated through inversion.
But inversion comes into its own when we are presented with a new, unique problem.
In such circumstances, average thinkers will impress upon themselves the importance of delivering everything required for a successful conclusion. Quite rightly. This is the natural way to respond to new challenges.
But great thinkers will force themselves to do the unnatural: they will add to their thinking armoury the things guaranteed to drive their challenge to disaster. And they will also recognise that the two things work in tandem. Inversion doesn’t replace conventional forward thinking; it complements it.
This sounds an obvious step, but when presented with real-world problems, it’s not an inevitability. By putting inversion as a theoretical tool at the front of our minds, we are much more likely to draw on it when it can produce the most value.
And here is the critical final point: sometimes avoiding stupidity is an easier long-term path to success than seeking brilliance. Reversing the way in which we look at problems can put this important point into perspective. Look no further than Charlie Munger, who has become something of a de facto champion for mental models, for an example of someone who extols the virtues of this approach in investment.
The Simple Steps of Inversion
As you have probably worked out by now, the art of inversion is to follow four simple steps:
Step One: Identify your goal.
Step Two: Identify issues that would prevent that goal from being realised
Step Three: Identify steps to avoid those issues
Step Four: Implement required actions
The next time you are solving a problem or looking for ideas, give inversion a shot. The results might surprise you.