In the 19th century, mathematician Carl Gustav Jacob Jacobi famously said:
Man muss immer umkehren.
which translates to “You must always invert.”
In linear algebra, proofs often require working with the inverse of a matrix to arrive at the solution. However, determining the inverse itself usually starts with the end goal in mind: verifying properties like or solving , the steps to constructing or applying the inverse become more focused and straightforward.
Inversion
- Complex problems are sometimes better solved backwards
- Invert the problem
- Prevention over pursuit
- Focusing on what to avoid rather than what to achieve
- Reverse Engineering
By identifying what we want to avoid, we often gain clearer insight into what we should actually do. The negative space defines the positive space.
Examples
- Investing:
- “How do I pick winning stocks?” ➞ “How do I avoid losing money?”
- “How much money can I make” ➞ “How can I lose not as much money?”
- Product Design: Instead of “What features should I add?”, ask “What would make users hate this product?”
- Health & Fitness: Instead of “How do I get in shape?”, ask “What habits would guarantee I stay unhealthy?”
- Decision-Making: Before committing to a major choice, run a premortem: imagine it is a year later and the decision has failed catastrophically — what went wrong?
| Instead of asking… | Ask… |
|---|---|
| How do I achieve X? | What would guarantee failure? |
| How do I win? | How do I ensure I don’t lose? |
| How do I get rich? | What would guarantee I go broke? |
| How do I be productive? | What would make me completely unproductive? |
| How do I build a great relationship? | What would ruin a relationship? |
| How do I find happiness? | What would make me miserable? |
| How do I stay healthy? | What would destroy my health? |