Exposition Problems

Recently, there was a blog post on my feed reader, referencing a recent paper on the topic of Open Exposition Problems in mathematics. To introduce this term, I'll use the same quote from the paper as was given in that blog post:

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.

This is an interesting idea, and I think it has applications in software development as well. The normal approach when explaining an algorithm is to just explain its steps. For any reasonably complex algorithm, it's also required to give some justification for why these steps achieve the desired result. Generally, the idea is that the student obtains enough of an understanding of the logic to produce a working version of the algorithm, and to extend it if need be.

That's fine, but the quoted text above goes further. It talks not only about the problem itself, but also about the motivation behind the steps of the solution, and about the student's feeling they could have constructed the solution themselves. This is something else entirely. We're now talking not just about explaining an algorithm, but explaining the process through which the algorithm was devised.

When writing code, we are always encouraged to add comments explaining how the code works. When the code needs to be maintained later, it's helpful to have these comments rather than having to work out what the code is doing. But, if someone's maintaining the code, it seems likely that they may be needing to write some similar code of their own. Maybe they need to extend this piece of code, or write a similar method in another language. In such a case, "exposition" comments might be useful as well, talking about how the code came about, other options that were rejected, and so on.

In any case, it's interesting to think about, both in terms of mathematics and software development. If nothing else, I learned the word perspicuous, and got a bit of a laugh that it was the word chosen to explain about making ideas completely clear.