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

The mathematics community currently views problem-solving and proof generation as indicators of a mathematician's true purpose: deepening human comprehension. When proofs can be produced without this comprehension, their significance as a proxy is called into question. This prompts a question: What alternative proxies should we adopt instead?

I propose that we establish a clear definition of a "motivated explanation" and recognize novel, compelling motivated explanations with academic credit comparable to the historical value of generating new proofs for open problems. Additionally, I believe this step is crucial for outsiders to better grasp what mathematicians contribute beyond proof generation.

While I admit to a personal bias stemming from my non-traditional career in math, focused on creating videos about the subject, my goal remains "furthering human understanding." My work emphasizes explanations and intuitive insights that resonate with the public, rather than solving outstanding problems. Despite this bias, I propose elevating the status of motivated explanations, not as popularization but as work that primarily aims to answer "how would you think of that?" even in fields requiring deep expertise.

Contrasting motivated explanations with proofs, the former begins with definitions in the middle, allowing new constructions only after the problem has been clearly established. In proofs, every statement must be correct, following as a necessary implication from previous statements. Motivated explanations, however, accept starting with an idea that may be incorrect but is relatable, often requiring correction.

One engaging genre of motivated explanation is "discovery fiction," a term coined by Michael Nielsen. It involves a narrative where an idea starts as a simple-but-wrong solution, breaks down, and is then fixed, leading to a new problem and solution. While proofs explain why a theorem is true, motivated explanations clarify why the theorem is the right one to pose and its context within the broader field.

A key shortcoming of motivated explanations is their subjective validity, unlike the binary nature of proofs. This subjectivity is why proof serves as a useful metric for progress. Shying away from subjective metrics may diminish the human aspects of the field. The word "motivated" is chosen over alternatives like "lucid" or "demystifying" because it allows for more verifiable assessment. Actionable guidelines include asking whether each new idea is clear in origin, allowing for a practical measure of completeness.

Examples of motivated explanations abound, such as the Princeton Companion to Mathematics, edited by Timothy Gowers. This anthology features contributions from experts in various fields, each introducing complex topics with clarity and intuition more typically found in personal conversations. Gowers, a Fields Medalist, discusses how this experience influenced his approach to editing the book, emphasizing the value of motivated explanations in mathematics.