Local vs. Global counterexamples

Lessons from mathematical proofs

The concept of local and global counterexamples comes from Imre Lakatos “Proofs and Refutations” which is a book about the process of mathematical discovery, including proofs. 

Two valuable concepts from Lakatos are Local and Global Counterexamples, defined below:

Local Counterexamples:

A local counterexample is an example that refutes a specific step within a larger proof or theorem, but does not necessarily invalidate the entire theorem. When faced with a local counterexample, mathematicians often try to modify or refine the specific step to accommodate the counterexample while preserving the overall structure of the proof.

Global Counterexamples

A global counterexample, on the other hand, refutes the entire theorem or conjecture. It demonstrates that the statement, as originally formulated, is false and requires significant revision or abandonment.

In the early days of a startup, your initial idea is a nothing more than a theorem. Your business is the proof.

As you refine and build, you are almost certain to encounter counterexamples that you hadn’t thought of in your original theorem. The value proposition doesn’t resonate for “XYZ customer type” or “Customers are only willing to buy when priced below ABC.” 

These are local counterexamples and they are ok. A local counterexample narrows your focus. Focus is good so therefore local counterexamples can be quite productive.

What is not good are global counterexamples. Global counterexamples indicate a much deeper problem in your idea or business model.

When pitching investors, they will often try to differentiate between these two types of counterexamples. Particularly if your idea has been tried before. 

Let’s take Instacart which drew comparison’s to Webvan when it first began. A lot of people at the time mistook Webvan’s failure for a global counterexample of the viability of grocery delivery. They assumed that because Webvan didn’t work, grocery delivery was fundamentally flawed.

What Apoorva Mehta and Instacart’s early supporters astutely realized was that this was not the case. Webvan’s demise could be attributed to a series of local counterexamples that, once remedied, would result in a viable business. These counterexamples included but are not limited to the gig economy, mobile phones, and the use of grocery stores versus owned inventory. As a ~$9b company at the time of this writing, it’s pretty clear that Apoorva and co were right. 

There’s a lot of alpha in identifying opportunities that have been written off as a result of prior failures being deemed global counterexamples. However, there’s also a lot of good money that can be spent after bad by failing to recognize the signs of a global counterexample when you see one. 

Weak and untested assumptions are often the common denominator between these failure modes. We’d all benefit from being a little more like a mathematician when testing our ideas.