Want To Deepen Your Thinking? Here Are a 19th Century Mathematician’s 2 Approaches For Thinking Through Problems.
Carl Jacobi was a 19th century mathematician who was famous for inventing the mental model: Inversion.
While he used this for mathematics, it has broad applications outside of math as well.
What’s so great about this principle is that it gives you tools to think about a problem from multiple angles.
It allows you to find vastly simpler ways of solving problems.
Here is a breakdown of the approaches to Inversion and how it applies.
Inversion involves thinking about the opposite of the end goal or starting point. You can do this in two different ways:
Approach #1: What do you want to avoid?
Instead of thinking about what you want to achieve, think about what you want to avoid.
Some classic examples of this could be:
- Instead of “How do I become rich?”, think “how do I not become poor?”
- Instead of “How do I make a smart decision?”, think “how do I not make a stupid decision?”
- Instead of “How can I be successful?”, think “how can I avoid failing?”
Asking yourself questions in these terms can help open up new avenues to problem solving you hadn’t thought of before.
Answering these questions (or others like it) will give you a list of things you SHOULDN’T do, increasing the likelihood you’ll succeed.
Approach #2: Work backwards
Instead of starting from the beginning of a scenario, think about the end and work your way backwards.
A practice that’s often used when working backwards is what’s sometimes referred to as a “Pre Mortem”.
When you start a project, imagine you’ve fast-forwarded six months in time and the project has failed.
Then tell the story of what went wrong.
The answer to this question will help you identify what you need to do in order to have a successful project.
Example of Applying Inversion
Let’s take an example and apply both approaches to it.
Imagine you are shopping for a house. You are in the early stages and just starting to look around.
Approach #1: What do you want to avoid?
What you may want to avoid while buying a house:
- Mortgage payments you can’t afford
- A house requiring too much work where it will be too expensive (or time consuming) to fix up
- A house without enough rooms to fit you and your family
- An ugly house architecture that doesn’t fit your tastes
These are some generic examples, and will obviously be way more specific to suit individual needs.
But what this exercise does is make it clear which houses you can immediately disqualify and not consider.
You may be house shopping and find a house with decor you fall in love with, but you can’t afford the mortgage payments.
This becomes an emotional decision that you may regret later. The above criteria will help you disqualify the house before you get too emotionally invested in it.
Approach #2: Work backwards
Imagine you bought a house that you regret. What went wrong?
- The mortgage payments were too expensive because the interest rates are too high
- The house needed way more work than expected—there was a plumbing issue the home inspection didn’t pick up
- The neighborhood is much noisier than expected
Now, what can you do preemptively to avoid these problems?
- Calculate the absolute maximum mortgage payments you can afford, inclusive of interest, real estate taxes, PMI, etc. Don’t deviate from that maximum no matter what the house looks like.
- Hire a plumber (who you trust or one recommended to you) to give an extra look at the house before going into contract.
- Visit the neighborhood at different times in the day to hear if things get noisier
The value of this exercise is to explore things that might become problematic later on that can be typically missed during the home buying process.
And more broadly, it allows you to approach a decision at more angles than you otherwise would have.