If math is more than proof, we need to better celebrate the rest of it

A growing sentiment within the mathematics community is that solving problems and creating proofs have traditionally been seen as proxies for the ultimate goal of mathematicians, which is to deepen human understanding. When proofs can be generated without that underlying understanding, their value as a proxy is diminished. This raises a question: what other proxies should be employed instead?

I propose we more clearly define and credit individuals who create "motivated explanations" similar to the historical acclaim given to mathematicians who generate new proofs of open problems. Additionally, I believe this step is crucial for helping non-mathematicians better understand the contributions mathematicians make.

To clarify, there is a distinction between a proof and a motivated explanation. In a proof, definitions are initially presented, followed by a series of logically correct statements derived from previous statements. In contrast, a motivated explanation begins with definitions in the middle. It is acceptable and desirable to start with an idea that may not be entirely correct, but its origins should be relatable and understandable.

One popular genre of motivated explanation is "discovery fiction," a term coined by Michael Nielsen. This involves developing an idea with a narrative that starts with a simple-but-wrong solution to a problem, identifying where it fails, and then discovering a new approach to tackle the problem. The scope of a proof is to explain why a particular theorem is true, while a motivated explanation aims not only to clarify why a theorem is true but also why that theorem is the right one to consider and how it fits into the broader context.

One challenge with motivated explanations is that their validity is not as binary as a proof's. This subjectivity is inherent because defining human understanding is inherently complex. While we might shy away from subjective metrics, embracing them allows us to acknowledge the more human aspects of the field. The term "motivated" is chosen over alternatives like "cleared" or "demystified" because it is more quantifiable.

While it may not be as rigidly verifiable as a proof, it is sufficient for practical purposes. In my own work, I often repeat the phrase, "I want this to feel like you could have discovered it yourself." This is not merely a placating statement but an actionable guideline to assess whether an explanation is complete or not, ensuring that each new idea introduced is clear and understandable, even if not entirely precise.